As per the given question,
Differentiate the following functions with respect to x: $\sin (1+{{x}^{2}}/1-{{x}^{2}})$
read more
Conic Sections
Find the equation of the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5, 0)
Given: Length of major axis is \[26\] and foci \[\left( \pm 5,\text{ }0 \right)\] Since the foci are on the x-axis, the major axis is along the x-axis. So, the equation of the ellipse will be of the...
Find the equation of each of the following parabolas (c) Focus at (–1, –2), directrix x – 2y + 3 = 0
Find the equation of each of the following parabolas (a) Directrix x = 0, focus at (6, 0) (b) Vertex at (0, 4), focus at (0, 2)
(a) The distance of any point on the parabola from its focus and its directrix is same. Given that, directrix, x = 0 and focus = (6, 0) If a parabola has a vertical axis, the standard form of the...
Find the equation of a circle passing through the point (7, 3) having radius 3 units and whose centre lies on the line y = x – 1.
the equation of a circle having centre (h, k), having radius as r units, is (x – h)2 + (y – k)2 = r2Centre lies on the line i.e., y = x – 1, Co – Ordinates are (h, k) = (h, h – 1)...
Find the equation of a circle of radius 5 which is touching another circle x2 + y2 – 2x – 4y – 20 = 0 at (5, 5).
Given \[\begin{array}{*{35}{l}} {{x}^{2}}~\text{ }-2x\text{ }+\text{ }{{y}^{2}}~\text{ }-4y\text{ }-\text{ }20\text{ }=\text{ }0 \\ {{x}^{2}}~\text{ }-2x\text{ }+\text{ }1\text{ }+{{y}^{2}}~\text{...
Find the equation of a circle whose centre is (3, –1) and which cuts off a chord of length 6 units on the line 2x – 5y + 18 = 0.
Using Pythagoras Theorem, (Hypotenuse)2 = (Base)2 + (Perpendicular)2 = (3)2 + (√29)2 = 29 + 9 = √38 Hypotenuse = √38 units (radius) Since, the radius bisects the chord into two equal halves, Since,...
Find the equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the straight line y – 4x + 3 = 0.
the equation of a circle having centre (h, k), having radius as r units, is \[{{\left( x\text{ }-\text{ }h \right)}^{2}}~+\text{ }{{\left( y\text{ }-\text{ }k \right)}^{2}}~=\text{ }{{r}^{2}}\ldots...
If the lines 2x – 3y = 5 and 3x – 4y = 7 are the diameters of a circle of area 154square units, then obtain the equation of the circle.
Since, diameters of a circle intersect at the centre of a circle, \[\begin{array}{*{35}{l}} 2x\text{ }-\text{ }3y\text{ }=\text{ }5\text{ }\ldots \ldots \ldots 1 \\ 3x\text{ }-\text{ }4y\text{...
Find the equation of the hyperbola with eccentricity 3/2 and foci at (± 2, 0).
Find the eccentricity of the hyperbola 9y^2 – 4x^2 = 36.
If the distance between the foci of a hyperbola is 16 and its eccentricity is √2, then obtain the equation of the hyperbola.
If the line y = mx + 1 is tangent to the parabola y2 = 4x then find the value of m.
equations are, y = mx + 1 & y2 = 4x By solving given equations we get (mx + 1)2 = 4x Expanding the above equation we get \[{{m}^{2}}{{x}^{2}}~+\text{ }2mx\text{ }+\text{ }1\text{ }=\text{ }4x\]...
If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.
Find the length of the line-segment joining the vertex of the parabola y2 = 4axand a point on the parabola where the line-segment makes an angle q to the x-axis.
Find the coordinates of a point on the parabola y2 = 8x whose focal distance is 4.
equation of an ellipse is y2 = 4ax, Also we have length of latus rectum = 4a Now by comparing the above two equations, 4a = 8 Therefore \[\begin{array}{*{35}{l}} a\text{ }=\text{ }2 \\...
Find the distance between the directrices of the ellipse x^2/36 + y^2/20 = 1
Find the equation of ellipse whose eccentricity is 2/3 , latus rectum is 5 and the centre is (0, 0).
If the eccentricity of an ellipse is 5/8 and the distance between its foci is 10, then find latus rectum of the ellipse.
Given the ellipse with equation 9x^2 + 25y^2 = 225, find the eccentricity and foci.
If the latus rectum of an ellipse is equal to half of minor axis, then find its eccentricity.
Find the equation of a circle concentric with the circle x^2 + y^2 – 6x + 12y + 15 = 0 and has double of its area.
Given equation of the circle is \[{{x}^{2}}~-\text{ }6x\text{ }+\text{ }{{y}^{2}}~+\text{ }12y\text{ }+\text{ }15\text{ }=\text{ }0\] The above equation can be written as \[\begin{array}{*{35}{l}}...
If the line y = √3x + k touches the circle x2 + y2 = 16, then find the value of k.
Find the equation of the circle having (1, –2) as its centre and passing through 3x + y = 14, 2x + 5y = 18
Solving the given equations, \[\begin{array}{*{35}{l}} 3x\text{ }+\text{ }y\text{ }=\text{ }14\text{ }\ldots \ldots \ldots .1 \\ 2x\text{ }+\text{ }5y\text{ }=\text{ }18\text{ }\ldots \ldots \ldots...
If one end of a diameter of the circle x^2 + y^2 – 4x – 6y + 11 = 0 is (3, 4), then find the coordinate of the other end of the diameter.
Given equation of the circle, \[\begin{array}{*{35}{l}} {{x}^{2}}~-\text{ }4x\text{ }+\text{ }{{y}^{2}}~\text{ }-6y\text{ }+\text{ }11\text{ }=\text{ }0 \\ {{x}^{2}}~\text{ }-4x\text{ }+\text{...
Find the equation of the circle which touches x-axis and whose centre is (1, 2).
Since the circle has a centre (1, 2) and also touches x-axis. Radius of the circle is, r = 2 The equation of a circle having centre (h, k), having radius as r units, is \[{{\left( x\text{ }-\text{...
If a circle passes through the point (0, 0) (a, 0), (0, b) then find the coordinates of its centre.
The equation of a circle having centre (h, k), having radius as r units, is \[{{\left( x\text{ }-\text{ }h \right)}^{2}}~+\text{ }{{\left( y\text{ }-\text{ }k \right)}^{2}}~=\text{ }{{r}^{2}}\]...
Show that the point (x,y) given by x= 2at/1+t^2 and y= a(1-t^2)/1+t^2 lies on a circle for all real values of t such that -1<=t<=1 where a is any given real number
Find the equation of the circle which touches the both axes in first quadrant and whose radius is a.
The circle touches both the x and y axes in the first quadrant and the radius is a. For a circle of radius a, the centre is (a, a). The equation of a circle having centre (h, k), having radius as r...
Find the equation for the ellipse that satisfies the given conditions: Major axis on the x-axis and passes through the points (4,3) and (6,2).
Solution:- Given: Major axis on the x-axis and passes through the points (4, 3) and (6, 2). Since the major axis is on the $x$-axis, the equation of the ellipse will be the form $\mathrm{x}^{2} /...
Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).
Given: Focus at\[\left( 0,\text{ }0 \right)\], significant pivot on the y-hub and goes through the focuses \[\left( 3,\text{ }2 \right)\text{ }and\text{ }\left( 1,\text{ }6 \right).\] Since the...
Find the equation for the ellipse that satisfies the given conditions: b = 3, c = 4, centre at the origin; foci on the x axis.
Given: \[b\text{ }=\text{ }3,\text{ }c\text{ }=\text{ }4,\]focus at the beginning and foci on the $x-axis$. Since the foci are on the $x-axis$, the significant hub is along the $x-axis$. In this...
Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), a = 4
Given: \[Foci\text{ }\left( \pm 3,\text{ }0 \right)\text{ }and\text{ }a\text{ }=\text{ }4\] Since the foci are on the, $x-axis$the significant pivot is along the$x-axis$. Thus, the condition of the...
Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6).
Given: Length of significant pivot is \[16\text{ }and\text{ }foci\text{ }\left( 0,~\pm 6 \right).\] Since the foci are on the$y-axis$, the significant hub is along the $y-axis.$ Along these lines,...
Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (0, ±√5), ends of minor axis (±1, 0)
Given: Closures of significant pivot \[\left( 0,~\pm \surd 5 \right)\]and finishes of minor hub \[\left( \pm 1,\text{ }0 \right)\] Here, the significant hub is along the$y-axis$. Thus, the condition...
Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (± 3, 0), ends of minor axis (0, ±2)
Given: Closures of significant pivot \[\left( \pm \text{ }3,\text{ }0 \right)\]and finishes of minor hub \[\left( 0,~\pm 2 \right)\] Here, the significant hub is along the\[x-pivot\]. Thus, the...
Find the equation for the ellipse that satisfies the given conditions: Vertices (± 6, 0), foci (± 4, 0)
Given: \[Vertices\text{ }\left( \pm ~6,\text{ }0 \right)\text{ }and\text{ }foci\text{ }\left( \pm \text{ }4,\text{ }0 \right)\] Here, the vertices are on the \[x-pivot.\] Along these lines, the...
Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ± 13), foci (0, ± 5)
Given: \[Vertices\text{ }\left( 0,~\pm ~13 \right)\text{ }and\text{ }foci\text{ }\left( 0,~\pm \text{ }5 \right)\] Here, the vertices are on the \[x-pivot.\] Along these lines, the condition of the...
Find the equation for the ellipse that satisfies the given conditions: Vertices (± 5, 0), foci (± 4, 0)
Given: \[Vertices\text{ }\left( \pm \text{ }5,\text{ }0 \right)\text{ }and\text{ }foci\text{ }\left( \pm \text{ }4,\text{ }0 \right)\] Here, the vertices are on the \[x-pivot.\] Along these lines,...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{9}{{\mathbf{y}}^{\mathbf{2}}}~=\text{ }\mathbf{36}\]
Given: The condition is \[4{{x}^{2}}~+\text{ }9{{y}^{2}}~=\text{ }36\text{ }or\text{ }{{x}^{2}}/9\text{ }+\text{ }{{y}^{2}}/4\text{ }=\text{ }1\text{ }or\text{ }{{x}^{2}}/{{3}^{2}}~+\text{...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[\mathbf{16}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }{{\mathbf{y}}^{\mathbf{2}}}~=\text{ }\mathbf{16}\]
Given: The condition is \[16{{x}^{2}}~+\text{ }{{y}^{2}}~=\text{ }16\text{ }or\text{ }{{x}^{2}}/1\text{ }+\text{ }{{y}^{2}}/16\text{ }=\text{ }1\text{ }or\text{ }{{x}^{2}}/{{1}^{2}}~+\text{...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[\mathbf{36}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}~=\text{ }\mathbf{144}\]
Given: The condition is \[36{{x}^{2}}~+\text{ }4{{y}^{2}}~=\text{ }144\text{ }or\text{ }{{x}^{2}}/4\text{ }+\text{ }{{y}^{2}}/36\text{ }=\text{ }1\text{ }or\text{ }{{x}^{2}}/{{2}^{2}}~+\text{...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[{{x}^{2}}/100\text{ }+\text{ }{{y}^{2}}/400\text{ }=\text{ }1\]
Given: The condition is \[{{x}^{2}}/100\text{ }+\text{ }{{y}^{2}}/400\text{ }=\text{ }1\] Here, the denominator of \[{{y}^{2}}/400\]is more noteworthy than the denominator of\[{{x}^{2}}/100\]. Thus,...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[{{\mathbf{x}}^{\mathbf{2}}}/\mathbf{49}\text{ }+\text{ }{{\mathbf{y}}^{\mathbf{2}}}/\mathbf{36}\text{ }=\text{ }\mathbf{1}\]
Given: The condition is \[{{x}^{2}}/49\text{ }+\text{ }{{y}^{2}}/36\text{ }=\text{ }1\] Here, the denominator of \[{{x}^{2}}/49\]is more noteworthy than the denominator of\[{{y}^{2}}/36\]. Thus, the...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[{{\mathbf{x}}^{\mathbf{2}}}/\mathbf{25}\text{ }+\text{ }{{\mathbf{y}}^{\mathbf{2}}}/\mathbf{100}\text{ }=\text{ }\mathbf{1}\]
Given: The condition is \[{{x}^{2}}/25\text{ }+\text{ }{{y}^{2}}/100\text{ }=\text{ }1\] Here, the denominator of \[{{y}^{2}}/100\] is more noteworthy than the denominator of\[{{x}^{2}}/25\]. Thus,...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[{{\mathbf{x}}^{\mathbf{2}}}/\mathbf{16}\text{ }+\text{ }{{\mathbf{y}}^{\mathbf{2}}}/\mathbf{9}\text{ }=\text{ }\mathbf{1}\]
Given: The condition is \[{{x}^{2}}/16\text{ }+\text{ }{{y}^{2}}/9\text{ }=\text{ }1\text{ }or\text{ }{{x}^{2}}/{{4}^{2}}~+\text{ }{{y}^{2}}/{{3}^{2}}~=\text{ }1\] Here, the denominator of...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[{{\mathbf{x}}^{\mathbf{2}}}/\mathbf{4}\text{ }+\text{ }{{\mathbf{y}}^{\mathbf{2}}}/\mathbf{25}\text{ }=\text{ }\mathbf{1}\]
Given: The condition is \[~{{x}^{2}}/4\text{ }+\text{ }{{y}^{2}}/25\text{ }=\text{ }1\] Here, the denominator of \[{{y}^{2}}/25\]is more noteworthy than the denominator of \[~{{x}^{2}}/4.\] Thus,...
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. \[{{\mathbf{x}}^{\mathbf{2}}}/\mathbf{36}\text{ }+\text{ }{{\mathbf{y}}^{\mathbf{2}}}/\mathbf{16}\text{ }=\text{ }\mathbf{1}\]
Given: The condition is \[{{x}^{2}}/36\text{ }+\text{ }{{y}^{2}}/16\text{ }=\text{ }1\] Here, the denominator of \[{{x}^{2}}/36\]is more greater than the denominator of \[{{y}^{2}}/16\] Thus, the...
Find the equations of the hyperbola satisfying the given conditions. Foci (0, ±√10), passing through (2, 3)
Given: Foci $(0, \pm \sqrt{10})$ and passing through $(2,3)$ Here, the foci are on $y$-axis. The eq. of the hyperbola is of the form $y^{2} / a^{2}-x^{2} / b^{2}=1$ Since, the foci are $(\pm...
Find the equations of the hyperbola satisfying the given conditions. Vertices (±7, 0), e = 4/3
Given: \[Vertices\text{ }\left( \pm 7,\text{ }0 \right)\text{ }and\text{ }e\text{ }=\text{ }4/3\] Here, the foci are on $x-axis.$ The condition of the hyperbola is of the structure...
Find the equations of the hyperbola satisfying the given conditions. Foci (± 4, 0), the latus rectum is of length 12
Given: Foci \[\left( \pm ~4,\text{ }0 \right)\]and the latus rectum is of length \[12\] Here, the foci are on $x-axis.$ The condition of the hyperbola is of the structure...
Find the equations of the hyperbola satisfying the given conditions. Foci (± 3√5, 0), the latus rectum is of length 8.
Given: Foci \[\left( \pm ~3\surd 5,\text{ }0 \right)\]and the latus rectum is of length \[8.\] Here, the foci are on \[x-axis\] The condition of the hyperbola is of the structure...
Find the equations of the hyperbola satisfying the given conditions. Foci (0, ±13), the conjugate axis is of length 24.
Given: Foci \[\left( 0,~\pm 13 \right)~\] and the cross over pivot is of length \[24.\] Here, the foci are on \[y-axis.\] The condition of the hyperbola is of the structure...
Find the equations of the hyperbola satisfying the given conditions Foci (±5, 0), the transverse axis is of length 8.
Given: Foci \[~\left( \pm 5,\text{ }0 \right)\]and the cross over pivot is of length \[8.\] Here, the foci are on \[x-axis.\] The condition of the hyperbola is of the structure...
Find the equations of the hyperbola satisfying the given conditions Vertices (0, ± 3), foci (0, ± 5)
Given: \[Vertices\text{ }\left( 0,~\pm ~3 \right)\text{ }and\text{ }foci\text{ }\left( 0,~\pm ~5 \right)\] Here, the vertices are on the$y-axis$. Along these lines, the condition of the hyperbola is...
Find the equations of the hyperbola satisfying the given conditions Vertices (0, ± 5), foci (0, ± 8)
Given: \[Vertices\text{ }\left( 0,~\pm ~5 \right)\text{ }and\text{ }foci\text{ }\left( 0,~\pm ~8 \right)\] Here, the vertices are on the $y-axis$ Along these lines, the condition of the hyperbola is...
Find the equations of the hyperbola satisfying the given conditions. Vertices (±2, 0), foci (±3, 0)
Given: \[Vertices\text{ }\left( \pm 2,\text{ }0 \right)\text{ }and\text{ }foci\text{ }\left( \pm 3,\text{ }0 \right)\] Here, the vertices are on the \[x-axis\] Along these lines, the condition of...
Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas . 49y^2 – 16x^2 = 784.
Given: The condition is \[49{{y}^{2}}~\text{ }16{{x}^{2}}~=\text{ }784.\] Let us divide the whole equation by \[784,\]we get \[\begin{align} & 49{{y}^{2}}/784\text{ }\text{...
Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. 5y^2 – 9x^2 = 36
Given: The condition is \[5{{y}^{2}}~\text{ }9{{x}^{2}}~=\text{ }36\] Let us divide the whole equation by \[36,\]we get \[\begin{array}{*{35}{l}} 5{{y}^{2}}/36\text{ }\text{ }9{{x}^{2}}/36\text{...
Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. 16x^2 – 9y^2 = 576
Given: The condition is \[16{{x}^{2}}~\text{ }9{{y}^{2}}~=\text{ }576\] Let us divide the whole equation by\[576\], we get \[\begin{array}{*{35}{l}} 16{{x}^{2}}/576\text{ }\text{...
Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. 9y^2 – 4x^2 = 36
Given: The condition is \[~9{{y}^{2}}~\text{ }4{{x}^{2}}~=\text{ }36\text{ }or\text{ }{{y}^{2}}/4\text{ }\text{ }{{x}^{2}}/9\text{ }=\text{ }1\text{ }or\text{ }{{y}^{2}}/{{2}^{2}}~\text{...
Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. y^2/9 – x^2/27 = 1
Given: The condition is \[{{y}^{2}}/9\text{ }\text{ }{{x}^{2}}/27\text{ }=\text{ }1\text{ }or\text{ }{{y}^{2}}/{{3}^{2}}~\text{ }{{x}^{2}}/{{27}^{2}}~=\text{ }1\] On contrasting this condition and...
Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. x^2/16 – y^2/9 = 1
The condition is \[~{{x}^{2}}/16\text{ }-\text{ }{{y}^{2}}/9\text{ }=\text{ }1\text{ }or\text{ }{{x}^{2}}/{{4}^{2}}~-\text{ }{{y}^{2}}/{{3}^{2}}~=\text{ }1\] On contrasting this condition and the...
An equilateral triangle is inscribed in the parabola y^2 = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
Allow us to consider \[OAB\]be the symmetrical triangle engraved in parabola\[{{y}^{2}}~=\text{ }4ax\]. Allow AB to cross the x – hub at point C. Diagrammatic portrayal of the circle is as per the...
A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man
Leave \[A\text{ }and\text{ }B\]alone the places of the two banner posts and \[P\left( x,\text{ }y \right)\]be the situation of the man. In this way, \[PA\text{ }+\text{ }PB\text{ }=\text{ }10.\] We...
Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus rectum.
The given parabola is \[{{x}^{2}}~=\text{ }12y.\] On contrasting this condition and\[{{x}^{2}}~=\text{ }4ay\], we get, \[\begin{array}{*{35}{l}} 4a\text{ }=\text{ }12 \\ a\text{ }=\text{ }12/4 \\...
A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.
Leave AB alone the pole making a point \[\theta with\text{ }OX\text{ }and\text{ }P\left( x,y \right)\]be the point on it to such an extent that \[AP\text{ }=\text{ }3cm.\] Diagrammatic portrayal is...
An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
Since, the tallness and width of the circular segment from the middle is \[2m\text{ }and\text{ }8m\]separately, obviously the length of the significant hub is\[8m\], while the length of the...
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
We realize that the vertex is at the absolute bottom of the link. The beginning of the facilitate plane is taken as the vertex of the parabola, while its upward hub is brought the positive \[y\text{...
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
We realize that the beginning of the facilitate plane is taken at the vertex of the curve, where its upward pivot is along the positive$y-axis.$ Diagrammatic portrayal is as per the following: The...
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
We realize that the beginning of the facilitate plane is taken at the vertex of the allegorical reflector, where the pivot of the reflector is along the positive \[x\text{ }\text{ axis}\]....
Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis.
We realize that the vertex is \[\left( 0,\text{ }0 \right)\]and the hub of the parabola is the \[y-axis\] The condition of the parabola is both of the from \[{{x}^{2~}}=\text{ }4ay\text{ }or\text{...
Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0) passing through (2, 3) and axis is along x-axis.
We realize that the vertex is \[\left( 0,\text{ }0 \right)\]and the hub of the parabola is the \[x-axis\] The condition of the parabola is both of the from \[~{{y}^{2~}}=\text{ }4ax\text{ }or\text{...
Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0); focus (–2, 0)
Given: \[Vertex\text{ }\left( 0,\text{ }0 \right)\text{ }and\text{ }focus\text{ }\left( -2,\text{ }0 \right)\] We realize that the vertex of the parabola is \[\left( 0,\text{ }0 \right)~\]and the...
Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0); focus (3, 0)
Given: Vertex \[\left( 0,\text{ }0 \right)\]and concentration \[\left( 3,\text{ }0 \right)\] We realize that the vertex of the parabola is \[\left( 0,\text{ }0 \right)\]and the attention lies on the...
Find the equation of the parabola that satisfies the given conditions: Focus (0,–3); directrix y = 3
Given: \[Focus\text{ }\left( 0,\text{ }-3 \right)\text{ }and\text{ }directrix\text{ }y\text{ }=\text{ }3\] We realize that the emphasis lies on the \[y-axis\] is the axis of the parabola. Along...
Find the equation of the parabola that satisfies the given conditions: Focus (6,0); directrix x = – 6
Given: \[Focus\text{ }\left( 6,0 \right)\text{ }and\text{ }directrix\text{ }x\text{ }=\text{ }-6\] We realize that the emphasis lies on the \[xaxis\] is the axis of the parabola. Along these lines,...
Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum. x^2 = – 9y
Given: The condition is \[{{x}^{2}}~=\text{ }-9y\] Here we realize that the coefficient of \[y\]is negative . Along these lines, the parabola opens towards downwards . On contrasting this condition...
Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum. y^2 = 10x
Given: The condition is \[{{y}^{2}}~=\text{ }10x\]. Here we realize that the coefficient of \[x\text{ }is\text{ }positive\] . Along these lines, the parabola opens towards right . On contrasting...
Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum. x^2 = – 16y
Given: The condition is \[{{x}^{2}}~=\text{ }-16y\]. Here, we realize that the coefficient of \[y\] is negative. Along these lines, the parabola opens towards downwards . On contrasting this...
Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum. y^2 = – 8x
Given: The condition is \[{{y}^{2}}~=\text{ }-8x\] Here we realize that the coefficient of $x$ is negative. Along these lines, the parabola opens towards left. On contrasting this condition and...
Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.x^2 = 6y
Given: The condition is \[{{x}^{2}}~=\text{ }6y\] Here we realize that the coefficient of $y$is positive . Along these lines, the parabola opens towards upwards . On contrasting this condition and...
Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum. y^2 = 12x
Given: The condition is \[{{y}^{2}}~=\text{ }12x\] Here we realize that the coefficient of $x$is positive. Along these lines, the parabola opens towards right . On contrasting this condition and...
Does the point (–2.5, 3.5) lie inside, outside or on the circle x2 + y2 = 25?
Given: The condition of the given circle is \[~{{x}^{2}}~+{{y}^{2}}~=\text{ }25.\] \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{ }{{y}^{2}}~=\text{ }25 \\ {{\left( x\text{ }\text{ }0...
Find the equation of a circle with centre (2,2) and passes through the point (4,5).
Given: The focal point of the circle is given as \[\left( h,\text{ }k \right)\text{ }=\text{ }\left( 2,2 \right)\] We realize that the circle goes through point \[\left( 4,5 \right),\]the range...
Find the equation of the circle passing through (0,0) and making intercepts a and b on the coordinate axes.
Allow us to consider the condition of the necessary circle be \[{{\left( x\text{ }\text{ }h \right)}^{2}}+\text{ }{{\left( y\text{ }\text{ }k \right)}^{2}}~={{r}^{2}}\] We realize that the circle...
Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2, 3).
Allow us to consider the condition of the necessary circle be \[{{\left( x\text{ }\text{ }h \right)}^{2}}+\text{ }{{\left( y\text{ }\text{ }k \right)}^{2}}~=\text{ }{{r}^{2}}\] We realize that the...
Find the equation of the circle passing through the points (2, 3) and (–1, 1) and whose centre is on the line x – 3y – 11 = 0.
Allow us to consider the condition of the necessary circle be \[{{\left( x\text{ }\text{ }h \right)}^{2~}}+\text{ }{{\left( y\text{ }\text{ }k \right)}^{2}}~=\text{ }{{r}^{2}}\] We realize that the...
Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x + y = 16.
Let us consider the condition of the necessary circle be \[{{\left( x\text{ }\text{ }h \right)}^{2}}+\text{ }{{\left( y\text{ }\text{ }k \right)}^{2}}~=\text{ }{{r}^{2}}\] We realize that the circle...
Find the centre and radius of the circles: 2x^2 + 2y^2 – x = 0
Given: The condition of the given circle is \[2{{x}^{2}}~+\text{ }2{{y}^{2}}~x\text{ }=\text{ }0.\] \[\begin{array}{*{35}{l}} 2{{x}^{2}}~+\text{ }2{{y}^{2}}~x\text{ }=\text{ }0 \\ \left(...
Find the centre and radius of the circles: x^2 + y^2 – 8x + 10y – 12 = 0
Given: The condition of the given circle is \[{{x}^{2}}~+\text{ }{{y}^{2}}~-8x\text{ }+\text{ }10y\text{ }-12\text{ }=\text{ }0.\] \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{ }{{y}^{2}}~\text{...
Find the centre and radius of the circles: x^2 + y^2 – 4x – 8y – 45 = 0
Given: The condition of the given circle is \[{{x}^{2}}~+\text{ }{{y}^{2}}~\text{ }4x\text{ }\text{ }8y\text{ }\text{ }45\text{ }=\text{ }0.\] \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{...
Find the centre and radius of the circles: (x + 5)^2 + (y – 3)^2 = 36
Given: The condition of the given circle is \[~{{\left( x\text{ }+\text{ }5 \right)}^{2}}~+\text{ }{{\left( y\text{ }\text{ }3 \right)}^{2}}~=\text{ }36\] \[{{\left( x\text{ }\text{ }\left( -5...
Find the equation of the circle with Centre (–a, –b) and radius √(a^2 – b^2)
Given: Focus \[\left( -a,\text{ }-b \right)\]and range \[\surd ({{a}^{2}}~\text{ }{{b}^{2}})\] Allow us to think about the situation of a circle with focus \[\left( h,\text{ }k \right)\]and Range...
Find the equation of the circle with Centre (1, 1) and radius √2
Given: Focus \[\left( 1,\text{ }1 \right)\] and range \[\surd 2\] Allow us to think about the situation of a circle with focus \[\left( h,\text{ }k \right)\]and Range\[~r\] is given as \[{{\left(...
Find the equation of the circle with Centre (1/2, 1/4) and radius (1/12)
Given: Focus \[\left( 1/2,\text{ }1/4 \right)~\]and range \[1/12\] Allow us to think about the situation of a circle with focus \[\left( h,\text{ }k \right)\]and Range\[~r\] is given as \[{{\left(...
Find the equation of the circle with Centre (–2, 3) and radius 4
Given: Focus \[~\left( -2,\text{ }3 \right)~\]and range \[4\] Allow us to think about the situation of a circle with focus \[\left( h,\text{ }k \right)\]and Range \[r\]is given as \[{{\left( x\text{...
Find the equation of the circle with Centre (0, 2) and radius 2
Given: Focus \[\left( 0,\text{ }2 \right)\]and range \[2\] Allow us to think about the situation of a circle with focus \[\left( h,\text{ }k \right)\]and Range \[r\]is given as Along these lines,...