differentiating the equation on both sides with respect to x, we get,
Find dy/dx in each of the following: $y^{3}-3 x y^{2}=x^{3}+3 x^{2} y$
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Areas Related to Circles
In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane
(a) (0, 0, 0) 3x – 4y + 12 z = 3
(b) (3, -2, 1) 2x – y + 2z + 3 = 0
Solution: (a) The distance of the point $(0,0,0)$ from the plane $3 x-4 y+12=3 \Rightarrow$ $3 x-4 y+12 z-3=0$ is $\begin{array}{l} \frac{\left|a x_{1}+b y_{1}+c...
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0
(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
Solution: (a) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ It is given that The eq. of the given planes are $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ Two planes are $\perp$ if the direction ratio of the...
Find the angle between the planes whose vector equations are $\vec{r}:(2 \hat{i}+2 \hat{j}-3 \hat{k})=5, \vec{r} \cdot(3 \hat{i}-3 \hat{j}+5 \hat{k})=3$
Solution: It is given that The eq. of the given planes are $\vec{r}(2 \hat{i}+2 \hat{j}-3 \hat{k})=5 \text { and } \vec{r}(3 \hat{i}-3 \hat{j}+5 \hat{k})=5$ If $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$...
Find the vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=7, \vec{r} \cdot(2 \hat{i}+5 \hat{j}+3 \hat{k})=9$ and through the point $(2,1,3)$
Solution: Let's consider the vector eq. of the plane passing through the intersection of the planes are $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=7...
Find the intercepts cut off by the plane 2x + y – z = 5.
Solution: It is given that The plane $2 x+y-z=5$ Let us express the equation of the plane in intercept form $x / a+y / b+z / c=1$ Where $a, b, c$ are the intercepts cut-off by the plane at $x, y$...
Find the vector and Cartesian equations of the planes
(a) that passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{i}+\hat{j}-\hat{k}$
(b) that passes through the point $(1,4,6)$ and the normal vector to the plane is $\hat{i}-2 \hat{j}+\hat{k}$
Solution: (a) That passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ Let's say that the position vector of the point $(1,0,-2)$...
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) x + y + z = 1
(b) 5y + 8 = 0
Solution: (a) $x+y+z=1$ Let the coordinate of the foot of $\perp \mathrm{P}$ from the origin to the given plane be $P(x, y, z)$ $x+y+z=1$ The direction ratio are $(1,1,1)$ $\begin{array}{l}...
Find the Cartesian equation of the following planes:
(a) $\overrightarrow{\mathrm{r}} \cdot[(\mathrm{s}-2 \mathrm{t}) \hat{\mathrm{i}}+(3-\mathrm{t}) \hat{\mathrm{j}}+(2 \mathrm{~s}+\mathrm{t}) \hat{\mathrm{k}}]=15$
Solution: Let $\overrightarrow{\mathrm{r}}$ be the position vector of $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is given by $\overrightarrow{\mathrm{r}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y}...
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$
Solution: It is given that, The vector $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$ Vector equation of the plane with position vector $\overrightarrow{\mathrm{r}}$ is $\vec{r} \cdot...
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) z = 2
(b) x + y + z = 1
Solution: (a) $z=2$ It is given that The eq. of the plane, $z=2$ or $0 x+0 y+z=2 \ldots (1) .$ The direction ratio of the normal $(0,0,1)$ Using the formula, $\begin{array}{l}...
On a square cardboard sheet of area \[784\] \[c{{m}^{2}}\], four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to two circular plates. Find the area of the square sheet not covered by the circular plates.
Given Area of the square = \[784\] \[c{{m}^{2}}\] Hence Side of the square = \[\sqrt{Area}\] = \[\sqrt{784}\] = \[28\] cm Given that the four circular plates are congruent, Therefore diameter of...
Four circular cardboard pieces of radii \[7\] cm are placed on a paper in such a way that each piece touches other two pieces. Find the area of the portion enclosed between these pieces.
solution From the given information, it is given that the four circles are placed such that each piece touches the other two pieces. Now by joining the centers of the circles by a line segment, we...
Find the area of the sector of a circle of radius \[5\] cm, if the corresponding arc length is \[3.5\] cm.
solution Given Radius of the circle = r = \[5\] cm Given Arc length of the sector = l = \[3.5\] cm Let us consider the central angle (in radians) be \[\theta \]. As we know that Arc length = Radius...
Three circles each of radius \[3.5\] cm are drawn in such a way that each of them touches the other two. Find the area enclosed between these circles.
Solution: Given that the three circles are drawn such that each of them touches the other two. Now, by joining the centers of the three circles, We get, AB = BC = CA = \[2\] (radius) = \[7\] cm...
In Fig. 11.17, ABCD is a trapezium with AB || DC, AB = \[18\] cm, DC = \[32\] cm and distance between AB and DC = \[14\] cm. If arcs of equal radii \[7\] cm with centres A, B, C and D have been drawn, then find the area of the shaded region of the figure.
Solution Given AB = \[18\] cm, DC = \[32\] cm Given, Distance between AB and DC = Height = \[14\] cm We know that Area of the trapezium = (\[1/2\]) × (Sum of parallel sides) × Height =...
A circular pond is \[17.5\] m is of diameter. It is surrounded by a \[2\] m wide path. Find the cost of constructing the path at the rate of Rs \[25\] per \[{{m}^{2}}\]
Solution: Given Diameter of the circular pond = \[17.5\] m Let us consider r be the radius of the park = \[(17.5/2)\] m = \[8.75\] m Given The circular pond is surrounded by a path of width \[2\] m....
Find the area of the segment of a circle of radius \[12\] m whose corresponding sector has a central angle of \[{{60}^{\circ }}\] (Use \[\pi =3.14\]).
Solution: From the given information, Radius of the circle = r = \[12\] cm ∴ OA = OB = \[12\] cm \[\angle AOB={{60}^{\circ }}\] (given) As triangle OAB is an isosceles triangle, ∴ \[\angle...
Find the area of the segment of a circle of radius \[12\] m whose corresponding sector has a central angle of \[{{60}^{\circ }}\] (Use \[\pi =3.14\]).
Solution: From the given information, Radius of the circle = r = \[12\] cm ∴ OA = OB = \[12\] cm \[\angle AOB={{60}^{\circ }}\] (given) As triangle OAB is an isosceles triangle, ∴ \[\angle...
Sides of a triangular field are \[15\] m, \[16\] m and \[17\] m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length \[7\] m each to graze in the field. Find the area of the field which cannot be grazed by the three animals.
Solution From the given question, We got Sides of the triangle are \[15\] m, \[16\] m and \[17\] m. Then, perimeter of the triangle = \[(15+16+17)\] m = \[48\]m Therefore, Semi-perimeter of the...
The diameters of front and rear wheels of a tractor are \[80\]cm and \[2\] m respectively. Find the number of revolutions that rear wheel will make in covering a distance in which the front wheel makes \[1400\] revolutions.
solution From the given question, We got, Diameter of front wheels = \[{{d}_{1}}\]= \[80\] cm we got, Diameter of rear wheels = \[{{d}_{2}}\]= \[2\]m = \[200\] cm Let us consider \[{{r}_{1}}\] be...
The area of a circular playground is \[22176\] \[{{m}^{2}}\]. Find the cost of fencing this ground at the rate of Rs \[50\] per metre.
From the given question, We got Area of the circular playground = \[22176\] \[{{m}^{2}}\] Let us consider r as the radius of the circle. Therefore, \[\pi {{r}^{2}}=22176\]...
In Fig. 11.7, AB is a diameter of the circle, \[AC=6\]cm and \[BC=8\] cm. Find the area of the shaded region (Use \[\pi =3.14\]).
Solution From the given question, \[AC=6\]cm and \[BC=8\] cm We know that a triangle in a semi-circle with hypotenuse as diameter is right angled triangle. By using Pythagoras theorem in triangle...
Find the area of the flower bed (with semi-circular ends) shown in Fig. 11.6.
Solution: From the given figure, We got that the Length and breadth of the rectangular portion (AFDC) of the flower bed are \[38\] cm and \[10\] cm respectively. We know that, Area of the flower bed...
A cow is tied with a rope of length \[14\] m at the corner of a rectangular field of dimensions \[20m\times 16m\]. Find the area of the field in which the cow can graze.
Let us consider ABCD be a rectangular field. Given, Length of the field = \[20\] m Given, Breadth of the field = \[16\] m From the given question, A cow is tied with a rope at a point A. Let us...
The wheel of a motor cycle is of radius \[35\] cm. How many revolutions per minute must the wheel make so as to keep a speed of \[66\] km/h?
From the question Radius of wheel = r = \[35\] cm We know that one revolution of the wheel is equal to Circumference of the wheel i.e., \[2\pi r\] = \[2\times (22/7)\times 35\] = \[220\] cm But,...
Find the area of a sector of a circle of radius \[28\]cm and central angle \[{{45}^{\circ }}\].
We know that Area of a sector of a circle = \[(1/2){{r}^{2}}\theta \], Here r is the radius and \[\theta \] is the angle in radians subtended by the arc at the center of the circle From the given...
In Fig. 11.5, a square of diagonal \[8\] cm is inscribed in a circle. Find the area of the shaded region.
Let us take a be the side of square. From the question we got, diagonal of square and diameter of circle is \[8\] cm In right angled triangle ABC, By Using Pythagoras theorem we got,...
Find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles of radii \[15\] cm and \[18\] cm.
Given Radius of first circle = \[{{r}_{1}}\] = \[15\] cm Given Radius of second circle = \[{{r}_{2}}\] = \[18\] cm Therefore, Circumference of first circle of radius \[{{r}_{1}}\]= \[2\pi...
In covering a distance s metres, a circular wheel of radius r metres makes \[s/2\pi r\] revolutions. Is this statement true? Why?
The given statement is True Explanation: The distance travelled by a circular wheel of radius r m in one revolution is equal to the circumference of the circle = \[2\pi r\] So we got, Number of...
Is it true that the distance travelled by a circular wheel of diameter d cm in one revolution is \[2d\] cm? Why?
The given statement is False Explanation: We know that, Circumference of the circle = \[2\pi d\](d is the diameter of the circle). Thus, the statement is false
Is it true to say that area of a segment of a circle is less than the area of its corresponding sector? Why?
The given statement is False Explanation: In major segment, area is not always greater than the are of its corresponding sector In minor segment, area is always greater than the area of its...
In Fig 11.3, a square is inscribed in a circle of diameter d and another square is circumscribing the circle. Is the area of the outer square four times the area of the inner square? Give reasons for your answer.
Solution: The given statement is False Explanation: From the fig, Let the Diameter of the circle = d Therefore, Diagonal of inner square (EFGH) = Side of the outer square (ABCD) = Diameter of circle...
Will it be true to say that the perimeter of a square circumscribing a circle of radius \[a\] cm is \[8a\]cm? Give reasons for your answer.
The given statement is true Explanation: Let \[r\] be the radius of circle and is equal \[a\] cm Therefore, Diameter of the circle = d = \[2\times Radius\] = \[2a\] cm From the question we got that...
Is the area of the circle inscribed in a square of side a cm, \[{{a}^{2}}\] \[c{{m}^{2}}\]? Give reasons for your answer.
The given statement is false Explanation: Let us assume a be the side of square. From the question we got that the circle is inscribed in the square. Therefore, Diameter of circle = Side of square...
If the perimeter of a circle is equal to that of a square, then the ratio of their areas is (A) \[22:7\] (B) \[14:11\] (C) \[7:22\] (D) \[11:14\]
The correct option is (B) \[14:11\] Explanation: Let us take r as the radius of the circle and a as the side of the square. From the given question, Perimeter of a circle of radius r = Perimeter of...
Area of the largest triangle that can be inscribed in a semi-circle of radius r units is (A) \[{{r}^{2}}\] sq. units (B) \[\frac{1}{2}{{r}^{2}}\] sq. units (C) \[2{{r}^{2}}\] sq. units (D) \[\sqrt{2}{{r}^{2}}\] sq. units
The correct option is (A) \[{{r}^{2}}\] sq. units Explanation: The largest triangle which can be inscribed in a semi-circle of radius r units is Base of triangle should be equal to the diameter of...
If the circumference of a circle and the perimeter of a square are equal, then (A) Area of the circle = Area of the square (B) Area of the circle > Area of the square (C) Area of the circle < Area of the square (D) Nothing definite can be said about the relation between the areas of the circle & square.
The correction option is (B) Area of the circle > Area of the square Explanation: From the given question, Circumference of a circle of radius r = Perimeter of a square of side a Let us take r...
If the sum of the circumferences of two circles with radii \[R1\] and \[R2\] is equal to the circumference of a circle of radius \[R\], then (A) \[{{R}_{1}}+{{R}_{2}}=R\] (B) \[{{R}_{1}}+{{R}_{2}}>R\] (C) \[{{R}_{1}}+{{R}_{2}}<R\] (D) Nothing definite can be said about the relation among \[{{R}_{1}}\], \[{{R}_{2}}\] & \[R\].
The Correct option(A) \[{{R}_{1}}+{{R}_{2}}=R\] Explanation: From the given question, We got sum of the circumferences of two circles with radii \[R1\] and \[R2\] is equal to the circumference of a...
If the sum of the areas of two circles with radii \[R1\] and \[R2\] is equal to the area of a circle of radius \[R\], then (A) \[{{R}_{1}}+{{R}_{2}}=R\] (B) \[R_{1}^{2}+R_{2}^{2}={{R}^{2}}\] (C) \[{{R}_{1}}+{{R}_{2}}<R\] (D) \[R_{1}^{2}+R_{2}^{2}<{{R}^{2}}\]
The Correct option is (B) \[R_{1}^{2}+R_{2}^{2}={{R}^{2}}\] Explanation: From the given question, We got sum of the areas of two circles with radii \[R1\] and \[R2\] is equal to the area of a circle...