#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 24 : The circle

#### Pages 21 - 22

Find the equation of the circle with:

Centre (−2, 3) and radius 4.

Find the equation of the circle with:

Centre (*a*, *b*) and radius\[\sqrt{a^2 + b^2}\]

Find the equation of the circle with:

Centre (0, −1) and radius 1.

Find the equation of the circle with:

Centre (*a* cos α, *a* sin α) and radius *a*.

Find the equation of the circle with:

Centre (*a*, *a*) and radius \[\sqrt{2}\]*a*.

Find the centre and radius of each of the following circles:

(*x* − 1)^{2} + *y*^{2} = 4

Find the centre and radius of each of the following circles:

(*x* + 5)^{2} + (*y* + 1)^{2} = 9

Find the centre and radius of each of the following circles:

*x*^{2} + *y*^{2} − 4*x* + 6*y* = 5

Find the centre and radius of each of the following circles:

*x*^{2} + *y*^{2}^{ }− *x* + 2*y* − 3 = 0.

Find the equation of the circle whose centre is (1, 2) and which passes through the point (4, 6).

Find the equation of the circle passing through the point of intersection of the lines *x* + 3*y* = 0 and 2*x* − 7*y* = 0 and whose centre is the point of intersection of the lines *x* + *y* + 1 = 0 and *x* − 2*y* + 4 = 0.

Find the equation of the circle whose centre lies on the positive direction of *y *- axis at a distance 6 from the origin and whose radius is 4.

If the equations of two diameters of a circle are 2*x* + *y* = 6 and 3*x* + 2*y* = 4 and the radius is 10, find the equation of the circle.

Find the equation of a circle

which touches both the axes at a distance of 6 units from the origin.

Find the equation of a circle

which touches *x*-axis at a distance 5 from the origin and radius 6 units.

Find the equation of a circle

which touches both the axes and passes through the point (2, 1).

Find the equation of a circle

passing through the origin, radius 17 and ordinate of the centre is −15.

Find the equation of the circle which has its centre at the point (3, 4) and touches the straight line 5*x* + 12*y* − 1 = 0.

Find the equation of the circle which touches the axes and whose centre lies on *x* − 2*y* = 3.

A circle whose centre is the point of intersection of the lines 2*x* − 3*y* + 4 = 0 and 3*x* + 4*y*− 5 = 0 passes through the origin. Find its equation.

A circle of radius 4 units touches the coordinate axes in the first quadrant. Find the equations of its images with respect to the line mirrors *x* = 0 and *y* = 0.

Find the equations of the circles touching *y*-axis at (0, 3) and making an intercept of 8 units on the *X*-axis.

Find the equations of the circles passing through two points on *Y*-axis at distances 3 from the origin and having radius 5.

If the lines 2*x* *−* 3*y* = 5 and 3*x −* 4*y* = 7 are the diameters of a circle of area 154 square units, then obtain the equation of the circle.

If the line *y* = \[\sqrt{3}\] *x* + *k* touches the circle *x*^{2} +* y*^{2} = 16, then find the value of *k*.

Find the equation of the circle having (1, −2) as its centre and passing through the intersection of the lines 3*x* + *y* = 14 and 2*x *+ 5*y* = 18.

If the lines 3*x* − 4*y* + 4 = 0 and 6*x* − 8*y* − 7 = 0 are tangents to a circle, then find the radius of the circle.

Show that the point (*x*, *y*) given by \[x = \frac{2at}{1 + t^2}\] and \[y = a\left( \frac{1 - t^2}{1 + t^2} \right)\] lies on a circle for all real values of *t* such that \[- 1 \leq t \leq 1\] where *a* is any given real number.

The circle *x*^{2} + *y*^{2} − 2*x* − 2*y* + 1 = 0 is rolled along the positive direction of *x*-axis and makes one complete roll. Find its equation in new-position.

One diameter of the circle circumscribing the rectangle *ABCD* is 4*y* = *x* + 7. If the coordinates of *A* and *B* are (−3, 4) and (5, 4) respectively, find the equation of the circle.

If the line 2*x* −* y *+ 1 = 0 touches the circle at the point (2, 5) and the centre of the circle lies on the line *x* + *y* − 9 = 0. Find the equation of the circle.

#### Pages 31 - 32

Find the coordinates of the centre and radius of each of the following circles: *x*^{2} + *y*^{2} + 6*x* − 8*y* − 24 = 0

Find the coordinates of the centre and radius of each of the following circles: 2*x*^{2} + 2*y*^{2} − 3*x* + 5*y* = 7

Find the coordinates of the centre and radius of each of the following circles: 1/2 (*x*^{2} + *y*^{2}) + *x* cos θ + *y* sin θ − 4 = 0

Find the coordinates of the centre and radius of each of the following circles: *x*^{2}^{ }+ *y*^{2} − *ax* − *by* = 0

Find the equation of the circle passing through the points:

(5, 7), (8, 1) and (1, 3)

Find the equation of the circle passing through the points:

(5, −8), (−2, 9) and (2, 1)

Find the equation of the circle passing through the points:

(0, 0), (−2, 1) and (−3, 2)

Find the equation of the circle which passes through (3, −2), (−2, 0) and has its centre on the line 2*x* − *y* = 3.

Find the equation of the circle which passes through the points (3, 7), (5, 5) and has its centre on the line *x* − 4*y* = 1.

Show that the points (3, −2), (1, 0), (−1, −2) and (1, −4) are concyclic.

Show that the points (5, 5), (6, 4), (−2, 4) and (7, 1) all lie on a circle, and find its equation, centre and radius.

Find the equation of the circle which circumscribes the triangle formed by the lines *x* + *y *+ 3 = 0, *x* − *y* + 1 = 0 and *x* = 3

Find the equation of the circle which circumscribes the triangle formed by the lines 2*x* + *y* − 3 = 0, *x* + *y* − 1 = 0 and 3*x* + 2*y* − 5 = 0

Find the equation of the circle which circumscribes the triangle formed by the lines

*x* + *y* = 2, 3*x* − 4*y* = 6 and *x* − *y* = 0.

Find the equation of the circle which circumscribes the triangle formed by the lines *y* = *x* + 2, 3*y* = 4*x* and 2*y* = 3*x*.

Prove that the centres of the three circles *x*^{2}^{ }+ *y*^{2} − 4*x* − 6*y* − 12 = 0, *x*^{2} + *y*^{2} + 2*x* + 4*y* − 10 = 0 and *x*^{2} + *y*^{2} − 10*x* − 16*y* − 1 = 0 are collinear.

Prove that the radii of the circles *x*^{2} + *y*^{2} = 1, *x*^{2} + *y*^{2} − 2*x* − 6*y* − 6 = 0 and *x*^{2} + *y*^{2} − 4*x* − 12*y* − 9 = 0 are in A.P.

Find the equation of the circle which passes through the origin and cuts off chords of lengths 4 and 6 on the positive side of the *x*-axis and *y*-axis respectively.

Find the equation of the circle concentric with the circle *x*^{2} + *y*^{2} − 6*x* + 12*y* + 15 = 0 and double of its area.

Find the equation to the circle which passes through the points (1, 1) (2, 2) and whose radius is 1. Show that there are two such circles.

Find the equation of the circle concentric with *x*^{2} + *y*^{2} − 4*x* − 6*y* − 3 = 0 and which touches the *y*-axis.

If a circle passes through the point (0, 0),(*a*, 0),(0, *b*) then find the coordinates of its centre.

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* − 4*x* + 3 = 0.

#### Pages 37 - 38

Find the equation of the circle, the end points of whose diameter are (2, −3) and (−2, 4). Find its centre and radius.

Find the equation of the circle the end points of whose diameter are the centres of the circles *x*^{2} + *y*^{2} + 6*x* − 14*y* − 1 = 0 and *x*^{2} + *y*^{2} − 4*x* + 10*y* − 2 = 0.

The sides of a square are *x* = 6, *x* = 9, *y* = 3 and *y* = 6. Find the equation of a circle drawn on the diagonal of the square as its diameter.

Find the equation of the circle circumscribing the rectangle whose sides are *x* − 3*y* = 4, 3*x* + *y* = 22, *x* − 3*y* = 14 and 3*x* + *y* = 62.

Find the equation of the circle passing through the origin and the points where the line 3*x* + 4*y* = 12 meets the axes of coordinates.

Find the equation of the circle which passes through the origin and cuts off intercepts *a*and *b* respectively from *x* and *y *- axes.

Find the equation of the circle whose diameter is the line segment joining (−4, 3) and (12, −1). Find also the intercept made by it on *y*-axis.

The abscissae of the two points *A* and *B* are the roots of the equation *x*^{2} + 2*ax* − *b*^{2} = 0 and their ordinates are the roots of the equation *x*^{2} + 2*px* − *q*^{2} = 0. Find the equation of the circle with *AB* as diameter. Also, find its radius.

*ABCD* is a square whose side is *a*; taking *AB* and *AD* as axes, prove that the equation of the circle circumscribing the square is *x*^{2} + *y*^{2} − *a* (*x* + *y*) = 0.

The line 2*x* − *y* + 6 = 0 meets the circle *x*^{2} + *y*^{2} − 2*y* − 9 = 0 at *A* and *B*. Find the equation of the circle on *AB* as diameter.

Find the equation of the circle which circumscribes the triangle formed by the lines *x* = 0, *y* = 0 and *lx* + *my* = 1.

Find the equations of the circles which pass through the origin and cut off equal chords of \[\sqrt{2}\] units from the lines *y* = *x* and *y* = − *x*.

#### Page 38

Write the length of the intercept made by the circle x^{2} + y^{2} + 2x − 4y − 5 = 0 on y-axis.

Write the coordinates of the centre of the circle passing through (0, 0), (4, 0) and (0, −6).

Write the area of the circle passing through (−2, 6) and having its centre at (1, 2).

If the abscissae and ordinates of two points *P* and *Q* are roots of the equations x^{2} + 2ax − b^{2} = 0 and x^{2} + 2px − q^{2} = 0 respectively, then write the equation of the circle with PQ as diameter.

Write the equation of the unit circle concentric with x^{2} + y^{2} − 8x + 4y − 8 = 0.

If the radius of the circle x^{2} + y^{2} + ax + (1 − a) y + 5 = 0 does not exceed 5, write the number of integral values a.

#### Page 38

Write the equation of the circle passing through (3, 4) and touching *y*-axis at the origin.

If the line y = mx does not intersect the circle (x + 10)^{2} + (y + 10)^{2} = 180, then write the set of values taken by m.

Write the coordinates of the centre of the circle inscribed in the square formed by the lines x = 2, x = 6, y = 5 and y = 9.

#### Pages 39 - 40

If the equation of a circle is λx^{2} + (2λ − 3) y^{2} − 4x + 6y − 1 = 0, then the coordinates of centre are

(4/3, −1)

(2/3, −1)

(−2/3, 1)

(2/3, 1)

If 2x^{2} + λxy + 2y^{2} + (λ − 4) x + 6y − 5 = 0 is the equation of a circle, then its radius is

\[3\sqrt{2}\]

\[2\sqrt{3}\]

\[2\sqrt{2}\]

none of these

The equation x^{2} + y^{2} + 2x − 4y + 5 = 0 represents

a point

a pair of straight lines

a circle of non-zero radius

none of these

If the equation (4*a* − 3) x^{2} + ay^{2} + 6x − 2y + 2 = 0 represents a circle, then its centre is

(3, −1)

(3, 1)

(−3, 1)

none of these

The radius of the circle represented by the equation 3x^{2} + 3y^{2} + λxy + 9x + (λ − 6) y + 3 = 0 is

\[\frac{3}{2}\]

\[\frac{\sqrt{17}}{2}\]

2/3

none of these

The number of integral values of λ for which the equation x^{2} + y^{2} + λx + (1 − λ) y + 5 = 0 is the equation of a circle whose radius cannot exceed 5, is

14

18

16

none of these

The equation of the circle passing through the point (1, 1) and having two diameters along the pair of lines x^{2} − y^{2} −2x + 4y − 3 = 0, is

x^{2} + y^{2} − 2x − 4y + 4 = 0

x^{2} + y^{2} + 2x + 4y − 4 = 0

x^{2} + y^{2} − 2x + 4y + 4 = 0

none of these

If the centroid of an equilateral triangle is (1, 1) and its one vertex is (−1, 2), then the equation of its circumcircle is

x^{2} + y^{2} − 2x − 2y − 3 = 0

x^{2} + y^{2} + 2x − 2y − 3 = 0

x^{2} + y^{2} + 2x + 2y − 3 = 0

none of these

If the point (2, k) lies outside the circles x^{2} + y^{2} + x − 2y − 14 = 0 and x^{2} + y^{2} = 13 then k lies in the interval

(−3, −2) ∪ (3, 4)

−3, 4

(−∞, −3) ∪ (4, ∞)

(−∞, −2) ∪ (3, ∞)

If the point (λ, λ + 1) lies inside the region bounded by the curve \[x = \sqrt{25 - y^2}\] and *y*-axis, then λ belongs to the interval

(−1, 3)

(−4, 3)

(−∞, −4) ∪ (3, ∞)

none of these

The equation of the incircle formed by the coordinate axes and the line 4x + 3y = 6 is

x^{2} + y^{2} − 6x −6y + 9 = 0

4 (x^{2} + y^{2} − x − y) + 1 = 0

4 (x^{2} + y^{2} + x + y) + 1 = 0

none of these

If the circles x^{2} + y^{2} = 9 and x^{2} + y^{2} + 8y + c = 0 touch each other, then c is equal to

15

-15

16

-16

If the circle x^{2} + y^{2} + 2ax + 8y + 16 = 0 touches x-axis, then the value of a is

± 16

±4

± 8

±1

The equation of a circle with radius 5 and touching both the coordinate axes is

x^{2} + y^{2} ± 10x ± 10y + 5 = 0

x^{2} + y^{2} ± 10x ± 10y = 0

x^{2}^{ }+ y^{2}^{ }± 10x ± 10y + 25 = 0

x^{2} + y^{2} ± 10x ± 10y + 51 = 0

The equation of the circle passing through the origin which cuts off intercept of length 6 and 8 from the axes is

x^{2} + y^{2} − 12x − 16y = 0

x^{2} + y^{2} + 12x + 16y = 0

x^{2} + y^{2} + 6x + 8y = 0

x^{2} + y^{2} − 6x − 8y = 0

The equation of the circle concentric with x^{2} + y^{2} − 3x + 4y − c = 0 and passing through (−1, −2) is

x^{2} + y^{2} − 3x + 4y − 1 = 0

x^{2} + y^{2} − 3x + 4y = 0

x^{2} + y^{2} − 3x + 4y + 2 = 0

none of these

The circle x^{2} + y^{2} + 2gx + 2fy + c = 0 does not intersect x-axis, if

g^{2} < c

g^{2} > c

g^{2} > 2c

none of these

The area of an equilateral triangle inscribed in the circle x^{2} + y^{2} − 6x − 8y − 25 = 0 is

\[\frac{225\sqrt{3}}{6}\]

25π

50π − 100

none of these

The equation of the circle which touches the axes of coordinates and the line \[\frac{x}{3} + \frac{y}{4} = 1\] and whose centres lie in the first quadrant is x^{2} + y^{2} − 2cx − 2cy + c^{2} = 0, where c is equal to

4

2

3

6

If the circles x^{2} + y^{2} = a and x^{2} + y^{2}^{ }− 6x − 8y + 9 = 0, touch externally, then a =

1

-1

21

16

If (x, 3) and (3, 5) are the extremities of a diameter of a circle with centre at (2, y), then the values of x and y are

(3, 1)

x = 4, y = 1

x = 8, y = 2

none of these

If (−3, 2) lies on the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 which is concentric with the circle x^{2} + y^{2} + 6x + 8y − 5 = 0, then c =

11

-11

24

none of these

Equation of the diameter of the circle x^{2} + y^{2} − 2x + 4y = 0 which passes through the origin is

x + 2y = 0

x − 2y = 0

2x + y = 0

2x − y = 0Let the diameter of the circle be *y* = *mx.*

Since the diameter of the circle passes through its centre, (1, −2) satisfies the equation of the diameter.

∴

Equation of the circle through origin which cuts intercepts of length a and b on axes is

x^{2} + y^{2} + ax + by = 0

x^{2} + y^{2} − ax − by = 0

x^{2} + y^{2} + bx + ay = 0

none of these

If the circles x^{2} + y^{2} + 2ax + c = 0 and x^{2} + y^{2} + 2by + c = 0 touch each other, then

\[\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c}\]

\[\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}\]

a + b = 2c

\[\frac{1}{a} + \frac{1}{b} = \frac{2}{c}\]

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 24 - The circle

RD Sharma solutions for Class 11 Maths chapter 24 (The circle) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 24 The circle are Standard Equation of a Circle, Latus Rectum, Standard Equation of Hyperbola, Eccentricity, Introduction of Hyperbola, Latus Rectum, Standard Equations of an Ellipse, Eccentricity, Special Cases of an Ellipse, Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus from the Centre of the Ellipse, Introduction of Ellipse, Latus Rectum, Standard Equations of Parabola, Introduction of Parabola, Concept of Circle, Sections of a Cone.

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