Straight Lines
If A(0, 0), b(2, 4) and C(6, 4) are the vertices of a ΔABC, find the equations of its sides.
Prove that the points A(1, 4), B(3, – 2) and C(4, – 5) are collinear. Also, find the equation of the line on which these points lie.
Find the equation of a line passing through the origin and making an angle of 1200 with the positive direction of the x – axis.
Find the equation of a line parallel to the y – axis at a distance of (i) 6 units to its right
(ii) 3 units to its left
(ii) 3 units to its left
Answer : (i) Equation of line parallel to y - axis is given by x = constant, as the x - coordinate of every point on the line parallel to y - axis is 6 i.e. constant. Now the point lies to the right...
Using slopes. Prove that the points A(-2, -1), B(1,0), C(4, 3) and D(1, 2) are the vertices of a parallelogram.
Find the value of x so that the line through (3, x) and (2, 7) is parallel to the line through (-1, 4) and (0, 6).
Find the slope of a line which passes through the points
(i) (0, 0) and (4, -2)
(ii) (0, -3) and (2, 1)
(iii) (2, 5) and (-4, -4)
(iv) (-2, 3) and (4, -6)
Find the values of k for which the line (k – 3) x – (4 – k2) y + k2 – 7k + 6 = 0 is (c) Passing through the origin.
(c) Here if the line is going through (0, 0) which is the beginning fulfills the given condition of line. \[\left( \mathbf{k}\text{ }\text{ }\mathbf{3} \right)\text{ }\left( \mathbf{0}...
The distance of the point of intersection of the lines 2x – 3y + 5 = 0 and 3x + 4y = 0 from the line 5x – 2y = 0 is
SOLUTION: The correct option is option(a) Explanation: Given two lines are \[\begin{array}{*{35}{l}} 2x\text{ }-\text{ }3y\text{ }+\text{ }5\text{ }=\text{ }0~\ldots \text{ }\left( i \right) \\...
If the line x/a+y/b=1 passes through the points (2,-3) and (4,-5) then (a,b) is: A. (1, 1) B. (– 1, 1) C. (1, – 1) D. (– 1, –1)
The correct option is option(D)- (– 1, –1)
The tangent of angles between the lines whose intercepts on the axes are a,-b and b, -a, respectively is: (A) a^2-b^2/ab (B) b^2-a^2/2 (C) b^2-a^2/2ab (D) none of these
The correct option is option(C)-
The equation of the line passing through the point (1, 2) and perpendicular to the line x + y + 1 = 0 is A. y – x + 1 = 0 B. y – x – 1 = 0 C. y – x + 2 = 0 D. y – x – 2 = 0
The correct option is option(B)- y – x – 1 = 0 line passing through the point (1, 2) And perpendicular to the line x + y + 1 = 0 Let the equation of line L is \[x\text{ }-\text{ }y\text{ }+\text{...
The equation of the straight line passing through the point (3, 2) and perpendicular to the line y = x is A. x – y = 5 B. x + y = 5 C. x + y = 1 D. x – y = 1
The correct option is option(B)- x + y = 5 straight line passing through the point (3, 2) And perpendicular to the line y = x Let the equation of line L be \[y\text{ }-\text{ }{{y}_{1}}~=\text{...
Slope of a line which cuts off intercepts of equal lengths on the axes is A. – 1 B. – 0 C. 2 D. √3
The correct option is option(A)- – 1
A line cutting off intercept – 3 from the y-axis and the tangent at angle to the x-axis is 3/5, its equation is
A. 5y – 3x + 15 = 0 B. 3y – 5x + 15 = 0 C. 5y – 3x – 15 = 0 D. None of these SOLUTION: The correct option is option(A)- 5y – 3x + 15 = 0
If p is the length of the perpendicular from origin on the line x/a=y/b=1 and a2,p2,and b2 ae in AP.Then show that a4+b4=0.
Solution: \[\begin{array}{*{35}{l}} \Rightarrow ~\left( {{a}^{2}}~+\text{ }{{b}^{2}} \right)\text{ }\left( {{a}^{2}}~+\text{ }{{b}^{2}} \right)\text{ }=\text{ }2\left( {{a}^{2}}{{b}^{2}} \right) \\...
p1 and p2 are points on either of the two lines y-√3 and |x|=2 at a distance of 5units from their point of intersection.Find the coordinates of the foot of perpendicular drawn from p1 and p2 on the bisector of theangle between the given lines.
Since, \[y\text{ }-\text{ }\surd 3\left| x \right|\text{ }=\text{ }2\]If x ≥ 0, then \[y\text{ }-\text{ }\surd 3x\text{ }=\text{ }2\text{ }\ldots ..\text{ }\left( i \right)\] If x < 0,...
If the sum of the distances of a moving point in a plane from the axes is 1, then find the locus of the point.
SOLUTION: Let the coordinates of point P be (a, b) Since, the sum of the distance from the axes to the point is always 1 \[\begin{array}{*{35}{l}} \therefore ~\left| x \right|\text{ }+\text{ }\left|...
Find the equations of the lines through the point of intersection of the lines x – y + 1 = 0 and 2x – 3y + 5 = 0 and whose distance from the point (3, 2) is 7/5.
Squaring both the sides, we get
Find the equation of the line which passes through the point (– 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5 : 3 by this point.
On cross multiplication we get \[\begin{array}{*{35}{l}} \Rightarrow ~-72x\text{ }+\text{ }160y\text{ }=\text{ }768 \\ \Rightarrow ~-36x\text{ }+\text{ }80y\text{ }=\text{ }384 \\ \Rightarrow...
In what direction should a line be drawn through the point (1, 2) so that its point of intersection with the line x + y = 4 is at a distance √6/3 from the given point.
A variable line passes through a fixed point P. The algebraic sum of the perpendiculars drawn from the points (2, 0), (0, 2) and (1, 1) on the line is zero. Find the coordinates of the point P.
\[\begin{array}{*{35}{l}} \Rightarrow ~2a\text{ }-\text{ }1\text{ }+\text{ }2b\text{ }-\text{ }1\text{ }+\text{ }a\text{ }+\text{ }b\text{ }-\text{ }1\text{ }=\text{ }0 \\ \Rightarrow ~3a\text{...
If the equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, – 1), then find the length of the side of the triangle.
Find the equation of one of the sides of an isosceles right angled triangle whose hypotenuse is given by 3x + 4y = 4 and the opposite vertex of the hypotenuse is (2, 2).
Find the equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of 120° with the positive direction of x-axis.
intercept of a line between the coordinate axes is divided by the point (–5, 4) in the ratio 1:2, then find the equation of the line.
For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes.
Find the equation of the line passing through the point of intersection of 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.
Given lines are \[\begin{array}{*{35}{l}} 2x\text{ }+\text{ }y\text{ }=\text{ }5\text{ }\ldots \ldots 1 \\ x\text{ }+\text{ }3y\text{ }=\text{ }-8\text{ }\ldots \ldots 2 \\ \end{array}\] Firstly,...
Find the equation of lines passing through (1, 2) and making angle 30° with y-axis.
SOLUTION: Given that line passing through (1, 2) making an angle 30° with y – axis. Angle made by the line with x – axis is (90° – 30°) = 60° ∴ Slope of the line, m = tan 60° = √3 So, the equation...
Show that the tangent of an angle between lines x/a+y/b=1 and x/a-y/b=1 is 2ab/a^2-b^2
Find the points on the line x + y = 4 which lie at a unit distance from the line 4x + 3y = 10.
Putting the value of y1 = 4 – x1 in equation 3, we get \[\begin{array}{*{35}{l}} 4{{x}_{1}}~+\text{ }3\left( 4\text{ }-\text{ }{{x}_{1}} \right)\text{ }=\text{ }5 \\ \Rightarrow ~4{{x}_{1}}~+\text{...
Find the equation of the lines which passes through the point (3, 4) and cuts off intercepts from the coordinate axes such that their sum is 14.
Find the angle between: y=2-√3(x+5) and y=2+√3(x-7)
Find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, – 1).
The points are A (5, 2), B (2, 3) and C (3, -1)
Find the equation of the straight line which passes through the point (1, – 2) and cuts off equal intercepts from axes.
A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.
GIVEN: $2x-3y+4=0$...(i) $3x+4y-5=0$...(ii) $6x-7y+8=0$...(iii) Here the individual is remaining at the intersection of the ways addressed by lines (1) and (2). By settling conditions (1) and (2) we...
Prove:
The product of the lengths of the perpendiculars drawn from the points to the line . GIVEN: IT CAN BE WRITTEN AS: \[bx\text{ }cos\text{ }\theta \text{ }+\text{ }ay\text{ }sin\text{ }\theta \text{...
A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.
Consider the directions of point An as (a, 0) Develop a line (AL) which is opposite to the x-pivot Here the point of occurrence is equivalent to point of reflection \[\angle BAL\text{ }=\angle...
Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0.
Here $9h+6k-7=3(3h+2k+6)$ or $9h+6k-7=3(3h+2k+6)$ $9h+6k-7=3(3h+2k+6)$ won’t be possible since $9h+6k-7=3(3h+2k+6)$ By additional estimation$7=18$ (isn't right) We realize that $9h+6k-7=-3(3h+2k+6)$...
If sum of the perpendicular distances of a variable point P (x, y) from the lines x + y – 5 = 0 and 3x – 2y + 7 = 0 is always 10. Show that P must move on a line.
SIMILARLY, equation of line for any signs of \[\left( x\text{ }+\text{ }y\text{ }\text{ }5 \right)\] and \[\left( 3x-\text{ }\text{ }2y\text{ }+\text{ }7 \right)\] CAN BE FOUND THUS, POINT P MUST...
If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.
GIVEN: \[y\text{ }=\text{ }3x\text{ }+\text{ }1\text{ }\ldots \text{ }\left( 1 \right)\] \[2y\text{ }=\text{ }x\text{ }+\text{ }3\text{ }\ldots \text{ }\left( 2 \right)\] \[y\text{ }=\text{...
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
Given: \[x\text{ }+\text{ }3y\text{ }=\text{ }7\text{ }\ldots \text{ }..\text{ }\left( 1 \right)\] Think about B (a, b) as the picture of point A (3, 8) So line (1) is opposite bisector of AB. On...
The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (−4, 1). Find the equation of the legs (perpendicular sides) of the triangle.
Consider ABC as the right points triangle where $\angle C={{90}^{\circ }}$ Here endlessness such lines are available. m is the slant of AC So the slant of $BC=-1/m$ Condition of AC $y-3=m(x-1)$ By...
Find the direction in which a straight line must be drawn through the point (–1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.
Think about \[y\text{ }=\text{ }mx\text{ }+\text{ }c\] as the line going through the point (- 1, 2) \[2\text{ }=\text{ }m\text{ }\left( -1 \right)\text{ }+\text{ }c\] So we get By additional...
Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.
given : \[2x\text{ }\text{ }y\text{ }=\text{ }0\text{ }\ldots \text{ }..\text{ }\left( 1 \right)\] \[4x\text{ }+\text{ }7y\text{ }+\text{ }5\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 2...
In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?
By cross augmentation \[\text{ }k\text{ }+\text{ }5\text{ }=\text{ }1\text{ }+\text{ }k\] We get \[2k\text{ }=\text{ }4\] \[k\text{ }=\text{ }2\] Henceforth, the line joining the...
Show that the equation of the line passing through the origin and making an angle θ with the line y = mx + c is
Think about \[y\text{ }=\text{ }m1x\] as the situation of the line going through the ORIGIN
Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes.
Consider the condition of the line having equivalent captures on the tomahawks as \[x/a\text{ }+\text{ }y/a\text{ }=\text{ }1\] It tends to be composed as \[x\text{ }+\text{ }y\text{ }=\text{...
Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x –2y = 3.
Think about m1 as the slant of the necessary line It tends to be composed as $$ \[y\text{ }=\text{ }1/2\text{ }x\text{ }\text{ }3/2\] which is of the structure \[y\text{ }=\text{ }mx\text{ }+\text{...
If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3 are concurrent, then show that m1 (c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0
It is provided that \[y\text{ }=\text{ }{{m}_{1}}x\text{ }+\text{ }{{c}_{1}}~\ldots ..\text{ }\left( 1 \right)\] \[y\text{ }=\text{ }m2x\text{ }+\text{ }c2\text{ }\ldots \text{ }..\text{ }\left( 2...
Find the value of p so that the three lines 3x + y – 2 = 0, px + 2y – 3 = 0 and 2x – y – 3 = 0 may intersect at one point.
It is given that \[3x\text{ }+\text{ }y\text{ }\text{ }2\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[px\text{ }+\text{ }2y\text{ }\text{ }3\text{ }=\text{ }0\text{ }\ldots \text{...
Find the area of the triangle formed by the lines y – x = 0, x + y = 0 and x – k = 0
It is given that \[y\text{ }\text{ }x\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[x\text{ }+\text{ }y\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)\] \[x\text{ }\text{...
Find the equation of a line drawn perpendicular to the line x/4 + y/6 = 1 through the point, where it meets the y-axis.
It is given that \[x/4\text{ }+\text{ }y/6\text{ }=\text{ }1\] We can compose it as \[3x\text{ }+\text{ }2y\text{ }\text{ }12\text{ }=\text{ }0\] So we get \[y\text{ }=\text{ }-\text{ }3/2\text{...
Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.
Here the situation of any line corresponding to the y-pivot is of the structure \[\mathbf{x}\text{ }=\text{ }\mathbf{a}\text{ }\ldots \text{ }.\text{ }\left( \mathbf{1} \right)\] Two given lines...
Find the perpendicular distance from the origin to the line joining the points :
according to the ques
What are the points on the y-axis whose distance from the line x/3 + y/4 = 1 is 4 units?
Consider (0, b) as the point on the y-pivot whose separation from line \[\mathbf{x}/\mathbf{3}\text{ }+\text{ }\mathbf{y}/\mathbf{4}\text{ }=\text{ }\mathbf{1}\text{ }\mathbf{is}\text{...
Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.
Consider the captures cut by the given lines on a and b axes \[\mathbf{a}\text{ }+\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{1}\text{ }\ldots \text{ }\left( \mathbf{1} \right)\]...
Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line √3x + y + 2 = 0.
according to the ques,
Find the values of k for which the line (k – 3) x – (4 – k2) y + k2 – 7k + 6 = 0 is (a) Parallel to the x-axis, (b) Parallel to the y-axis,
(a) Here if the line is corresponding to the x-pivot Slant of the line = Slope of the x-pivot It very well may be composed as \[\left( \mathbf{4}\text{ }\text{ }\mathbf{k2} \right)\text{...
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that 1/p2 = 1/a2 + 1/b2
Condition of a line whose catches on the axes are an and b is \[\mathbf{x}/\mathbf{a}\text{ }+\text{ }\mathbf{y}/\mathbf{b}\text{ }=\text{ }\mathbf{1}\] \[\mathbf{bx}\text{ }+\text{...
In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.
Leave AD alone the height of triangle ABC from vertex A. In this way, AD is opposite to BC Given: Vertices \[\mathbf{A}\text{ }\left( \mathbf{2},\text{ }\mathbf{3} \right),\text{...
If p and q are the lengths of perpendiculars from the origin to the lines x cos θ − y sin θ = k cos 2θ and x sec θ + y cosec θ = k, respectively, prove that p2 + 4q2 = k2
Given: The equations of given lines are x cos θ – y sin θ = k cos 2θ …………………… (1) x sec θ + y cosec θ = k ……………….… (2) Perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is...
The perpendicular from the origin to the line y = mx + c meets it at the point (–1, 2). Find the values of m and c.
Given: The opposite from the beginning meets the given line at \[\left( \text{ }\mathbf{1},\text{ }\mathbf{2} \right).\] The condition of line is \[\mathbf{y}\text{ }=\text{ }\mathbf{mx}\text{...
Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.
Allow us to consider the co-ordinates of the foot of the opposite from (- 1, 3) to the line \[\mathbf{3x}\text{ }\text{ }\mathbf{4y}\text{ }\text{ }\mathbf{16}\text{ }=\text{ }\mathbf{0}\text{...
Find the equation of the right bisector of the line segment joining the points (3, 4) and (–1, 2).
Given: The right bisector of a line section divides the line fragment at \[\mathbf{90}{}^\circ .\] End-points of the line portion AB are given as \[\mathbf{A}\text{ }\left( \mathbf{3},\text{...
Two lines passing through the point (2, 3) intersects each other at an angle of 60o. If slope of one line is 2, find equation of the other line.
Given\[:\text{ }\mathbf{m1}\text{ }=\text{ }\mathbf{2}\] Leave the slant of the principal line alone m1 What's more, left the incline of the other line alone m2. Point between the two lines is...
Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x – x1) + B (y – y1) = 0.
Let the slant of line \[\mathbf{Ax}\text{ }+\text{ }\mathbf{By}\text{ }+\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{0}\] be m \[\mathbf{Ax}\text{ }+\text{ }\mathbf{By}\text{ }+\text{...
The line through the points (h, 3) and (4, 1) intersects the line 7x − 9y −19 = 0. At right angle. Find the value of h.
Let the incline of the line going through (h, 3) and (4, 1) be m1 Then, at that point, \[\mathbf{m1}\text{ }=\text{ }\left( \mathbf{1}-\mathbf{3} \right)/\left( \mathbf{4}-\mathbf{h} \right)\text{...
Find angles between the lines √3x + y = 1 and x + √3y = 1.
Given: The lines are \[\surd \mathbf{3x}\text{ }+\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{x}\text{ }+\text{ }\surd \mathbf{3y}\text{ }=\text{ }\mathbf{1}\]...
Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3.
Given: The condition of line is \[\mathbf{x}\text{ }\text{ }\mathbf{7y}\text{ }+\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{0}\] Thus, \[\mathbf{y}\text{ }=\text{ }\mathbf{1}/\mathbf{7x}\text{...
Find equation of the line parallel to the line 3x − 4y + 2 = 0 and passing through the point (–2, 3).
Given: The line is \[\mathbf{3x}\text{ }\text{ }\mathbf{-4y}\text{ }+\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{0}\] In this way, \[\mathbf{y}\text{ }=\text{ }\mathbf{3x}/\mathbf{4}\text{ }+\text{...
Find the distance between parallel lines (i) 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0 (ii) l(x + y) + p = 0 and l (x + y) – r = 0
\[\left( \mathbf{i} \right)\text{ }\mathbf{15x}\text{ }+\text{ }\mathbf{8y}\text{ }\text{ }\mathbf{34}\text{ }=\text{ }\mathbf{0}\text{ }\mathbf{and}\text{ }\mathbf{15x}\text{ }+\text{...
Find the points on the x-axis, whose distances from the line x/3 + y/4 = 1 are 4 units.
Given: The condition of line is \[\mathbf{x}/\mathbf{3}\text{ }+\text{ }\mathbf{y}/\mathbf{4}\text{ }=\text{ }\mathbf{1}\] \[\mathbf{4x}\text{ }+\text{ }\mathbf{3y}\text{ }=\text{ }\mathbf{12}\]...
Find the distance of the point (–1, 1) from the line 12(x + 6) = 5(y – 2).
Given: The condition of the line is \[\mathbf{12}\left( \mathbf{x}\text{ }+\text{ }\mathbf{6} \right)\text{ }=\text{ }\mathbf{5}\left( \mathbf{y}\text{ }\text{ }\mathbf{2} \right).\]...
Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis. (iii) x – y = 4
\[\left( \mathbf{iii} \right)\text{ }\mathbf{x-}\text{ }\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{4}\] Given: The condition is \[\mathbf{x-}\text{ }\text{ }\mathbf{y}\text{ }+\text{...
Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis. (i) x – √3y + 8 = 0 (ii) y – 2 = 0
(i) x – √3y + 8 = 0 Given: The condition is \[\mathbf{x}\text{ }\text{ }\surd \mathbf{3y}\text{ }+\text{ }\mathbf{8}\text{ }=\text{ }\mathbf{0}\] Condition of line in typical structure is given...
Reduce the following equations into intercept form and find their intercepts on the axes. (iii) 3y + 2 = 0
\[\left( \mathbf{iii} \right)\text{ }\mathbf{3y}\text{ }+\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{0}\] Given: The condition is \[\mathbf{3y}\text{ }+\text{ }\mathbf{2}\text{ }=\text{...
Reduce the following equations into intercept form and find their intercepts on the axes. (i) 3x + 2y – 12 = 0 (ii) 4x – 3y = 6
\[\left( \mathbf{i} \right)\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{2y}\text{ }\text{ }\mathbf{-12}\text{ }=\text{ }\mathbf{0}\] Given: The condition is \[\mathbf{3x}\text{ }+\text{...
Reduce the following equations into slope – intercept form and find their slopes and the y – intercepts.(iii) y = 0
\[\left( \mathbf{iii} \right)\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{0}\] Given: The condition is \[\mathbf{y}\text{ }=\text{ }\mathbf{0}\] Slant – catch structure is given...
Reduce the following equations into slope – intercept form and find their slopes and the y – intercepts. (i) x + 7y = 0 (ii) 6x + 3y – 5 = 0
As indicated by the inquiry, \[~\left( \mathbf{i} \right)\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{7y}\text{ }=\text{ }\mathbf{0}\] Given: The condition is \[\mathbf{x}\text{ }+\text{...
By using the concept of equation of a line, prove that the three points (3, 0), (– 2, – 2) and (8, 2) are collinear
On the off chance that we need to demonstrate that the given three focuses \[\left( \mathbf{3},\text{ }\mathbf{0} \right),\text{ }\left( \text{ }\mathbf{2},\text{ }\text{ }\mathbf{2} \right)\text{...
Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find the equation of the line.
Allow us to consider, AB be the line portion to such an extent that r (h, k) separates it in the proportion \[\mathbf{1}:\text{ }\mathbf{2}.\] So the directions of An and B be (0, y) and (x, 0)...
P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is x/a + y/b = 2
Leave AB alone a line section whose midpoint is P (a, b). Leave the directions of An and B alone (0, y) and (x, 0) individually. \[\mathbf{a}\text{ }\left( \mathbf{y}\text{ }\text{ }\mathbf{2b}...
find the equation of the line which satisfy the given condition: The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs. 14/litre and 1220 litres of milk each week at Rs. 16/litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs. 17/litre?
Accepting the connection between selling cost and request is direct. Allow us to accept selling cost per liter along X-pivot and request along Y-hub, we have two focuses \[\left(...
find the equation of the line which satisfy the given condition: The length L (in centimetre) of a copper rod is a linear function of its Celsius temperature C. In an experiment, if L = 124.942 when C = 20 and L= 125.134 when C = 110, express L in terms of C.
Allow us to expect 'L' along X axis and 'C' along Y axis, we have two focuses \[\left( \mathbf{124}.\mathbf{942},\text{ }\mathbf{20} \right)\text{ }\mathbf{and}\text{ }\left(...
find the equation of the line which satisfy the given condition: The perpendicular from the origin to a line meets it at the point (–2, 9), find the equation of the line.
Given: Points are origin (0, 0) and (-2, 9). We know that slope, m = (y2 – y1)/(x2 – x1) = (9 – 0)/(-2-0) = -9/2 We realize that two non-vertical lines are opposite to one another if and provided...
find the equation of the line which satisfy the given condition: Find equation of the line through the point (0, 2) making an angle 2π/3 with the positive x-axis. Also, find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin.
Given: \[\mathbf{Point}\text{ }\left( \mathbf{0},\text{ }\mathbf{2} \right)\] and \[\mathbf{\theta }\text{ }=\text{ }\mathbf{2\pi }/\mathbf{3}\] We realize that \[\mathbf{m}\text{ }=\text{...
find the equation of the line which satisfy the given condition: Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9
We realize that condition of the line making blocks an and b on x-and y-axis, individually, is \[\mathbf{x}/\mathbf{a}\text{ }+\text{ }\mathbf{y}/\mathbf{b}\text{ }=\text{ }\mathbf{1}\text{ }.\text{...
find the equation of the line which satisfy the given condition: Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).
Given: the line cuts off equivalent captures on the organize tomahawks for example \[\mathbf{a}\text{ }=\text{ }\mathbf{b}.\] We realize that condition of the line blocks an and b on x-and y-pivot,...
find the equation of the line which satisfy the given condition: A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line
We realize that the directions of a point separating the line portion joining the focuses (x1, y1) and (x2, y2) inside in the proportion m: n are We know that slope, m = (y2 – y1)/(x2 – x1) = (3 –...
find the equation of the line which satisfy the given condition: Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6).
Given: Focuses are \[\left( \mathbf{2},\text{ }\mathbf{5} \right)\] and \[\left( -\text{ }\mathbf{3},\text{ }\mathbf{6} \right).\] We realize that slant, \[\mathbf{m}\text{ }=\text{ }\left(...
find the equation of the line which satisfy the given condition: The vertices of ΔPQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the median through the vertex R.
Given: Vertices of ΔPQR for example \[\mathbf{P}\text{ }\left( \mathbf{2},\text{ }\mathbf{1} \right),\text{ }\mathbf{Q}\text{ }\left( -\text{ }\mathbf{2},\text{ }\mathbf{3} \right)\text{...
find the equation of the line which satisfy the given condition: Perpendicular distance from the origin is 5 units and the angle made by the perpendicular with the positive x-axis is 30o.
Given: \[\mathbf{p}\text{ }=\text{ }\mathbf{5}\] and \[\mathbf{\omega }\text{ }=\text{ }\mathbf{30}{}^\circ \] We realize that the condition of the line having typical distance p from the...
find the equation of the line which satisfy the given condition: Passing through the points (–1, 1) and (2, – 4).
Given: Focuses \[\left( -\text{ }\mathbf{1},\text{ }\mathbf{1} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{2},\text{ }-\text{ }\mathbf{4} \right)\] We realize that the condition of the...
find the equation of the line which satisfy the given condition: Intersecting the y-axis at a distance of 2 units above the origin and making an angle of 30o with positive direction of the x-axis.
Given: \[\mathbf{\theta }\text{ }=\text{ }\mathbf{30}{}^\circ \] We realize that incline, \[\mathbf{m}\text{ }=\text{ }\mathbf{tan}\text{ }\mathbf{\theta }\] \[\mathbf{m}\text{ }=\text{...
find the equation of the line which satisfy the given condition: Intersecting the x-axis at a distance of 3 units to the left of origin with slope –2.
Given: Incline, \[\mathbf{m}\text{ }=\text{ }-\text{ }\mathbf{2}\] We realize that assuming a line L with incline m makes x-capture d, condition of L is \[\mathbf{y}\text{ }=\text{...
find the equation of the line which satisfy the given condition: Passing through (2, 2√3) and inclined with the x-axis at an angle of 75o.
Given: point \[\left( \mathbf{2},\text{ }\mathbf{2}\surd \mathbf{3} \right)\] and \[\mathbf{\theta }\text{ }=\text{ }\mathbf{75}{}^\circ \] Condition of line: \[\left( \mathbf{y}\text{ }\text{...
find the equation of the line which satisfy the given condition: Passing through (0, 0) with slope m.
Given: Point (0, 0) and slant, \[\mathbf{m}\text{ }=\text{ }\mathbf{m}\] We realize that the point (x, y) lies on the line with slant m through the decent point (x0, y0), if and provided that,...
find the equation of the line which satisfy the given condition: Passing through the point (– 4, 3) with slope 1/2
Given: Point (- 4, 3) and incline, \[\mathbf{m}\text{ }=\text{ }\mathbf{1}/\mathbf{2}\] We realize that the point (x, y) lies on the line with incline m through the decent point (x0, y0), if and...
find the equation of the line which satisfy the given condition: Write the equations for the x-and y-axes.
The y-arrangement of each point on x-axis is 0. ∴ Equation of x-axis is$$ \[\mathbf{y}\text{ }=\text{ }\mathbf{0}\] . The x-arrangement of each point on y-axis is 0. ∴ Equation of y-axis is...
Consider the following population and year graph (Fig 10.10), find the slope of the line AB and using it, find what will be the population in the year 2010?
We realize that, the line AB goes through focuses\[\mathbf{A}\text{ }\left( \mathbf{1985},\text{ }\mathbf{92} \right)\text{ }\mathbf{and}\text{ }\mathbf{B}\text{ }\left( \mathbf{1995},\text{...
If three points (h, 0), (a, b) and (0, k) lie on a line, show that a/h + b/k = 1
Allow us to consider if the given focuses \[\mathbf{A}\text{ }\left( \mathbf{h},\text{ }\mathbf{0} \right),\text{ }\mathbf{B}\text{ }\left( \mathbf{a},\text{ }\mathbf{b} \right)\text{...
A line passes through (x1, y1) and (h, k). If slope of the line is m, show that k – y1 = m (h – x1).
Given: the slant of the line is 'm' The slant of the line going through \[\left( \mathbf{x1},\text{ }\mathbf{y1} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{h},\text{ }\mathbf{k} \right)\]...
The slope of a line is double of the slope of another line. If tangent of the angle between them is 1/3, find the slopes of the lines.
Allow us to consider 'm1' and 'm' be the incline of the two given lines to such an extent that \[\mathbf{m1}\text{ }=\text{ }\mathbf{2m}\] We realize that in case θ is the point somewhere within l1...
Find the angle between the x-axis and the line joining the points (3, –1) and (4, –2).
Slope of the line joining the focuses \[\left( \mathbf{3},\text{ }-\text{ }\mathbf{1} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{4},\text{ }-\text{ }\mathbf{2} \right)\] is given by ...
Without using distance formula, show that points (– 2, – 1), (4, 0), (3, 3) and (–3, 2) are the vertices of a parallelogram
Leave the given point alone \[\mathbf{A}\text{ }\left( -\text{ }\mathbf{2},\text{ }-\text{ }\mathbf{1} \right)\text{ },\text{ }\mathbf{B}\text{ }\left( \mathbf{4},\text{ }\mathbf{0} \right)\text{...
Find the value of x for which the points (x, – 1), (2, 1) and (4, 5) are collinear.
On the off chance that the focuses \[\left( \mathbf{x},\text{ }\text{ }\mathbf{1} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{1} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{4},\text{...
Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
We realize that, if a line makes a point of \[\mathbf{30}{}^\circ \] with the positive bearing of y-pivot estimated against clock-wise , then, at that point, the point made by the line with the...
Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right-angled triangle.
The vertices of the given triangle are \[\left( \mathbf{4},\text{ }\mathbf{4} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{5} \right)\text{ }\mathbf{and}\text{ }\left( \text{ }\mathbf{1},\text{...
Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, – 4) and B (8, 0)
The co-ordinates of mid-point of the line portion joining the focuses \[\mathbf{P}\text{ }\left( \mathbf{0},\text{ }\text{ }\mathbf{4} \right)\text{ }\mathbf{and}\text{ }\mathbf{B}\text{ }\left(...
Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
Allow us to consider \[\left( \mathbf{a},\text{ }\mathbf{0} \right)\] be the point on the x-hub that is equidistant from the point \[\left( \mathbf{7},\text{ }\mathbf{6} \right)\] and \[\left(...
Find the distance between P (x1, y1) and Q (x2, y2) when: (i) PQ is parallel to the y-axis, (ii) PQ is parallel to the x-axis
Given: Focuses \[\mathbf{P}\text{ }\left( \mathbf{x1},\text{ }\mathbf{y1} \right)\text{ }\mathbf{and}\text{ }\mathbf{Q}\left( \mathbf{x2},\text{ }\mathbf{y2} \right)\] (i) When PQ is corresponding...
The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle
Allow us to consider ABC be the given symmetrical triangle with side 2a. Where, \[\mathbf{AB}\text{ }=\text{ }\mathbf{BC}\text{ }=\text{ }\mathbf{AC}\text{ }=\text{ }\mathbf{2a}\] In the above...
Draw a quadrilateral in the Cartesian plane, whose vertices are (– 4, 5), (0, 7), (5, – 5) and (– 4, –2). Also, find its area.
Leave ABCD alone the given quadrilateral with vertices \[\mathbf{A}\text{ }\left( -\text{ }\mathbf{4},\mathbf{5} \right)\text{ },\text{ }\mathbf{B}\text{ }\left( \mathbf{0},\mathbf{7} \right),\text{...