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

Mathematics Class 11

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RD Sharma Mathematics Class 11

Mathematics Class 11

Chapter 24: The circle

Ex. 24.10Ex. 24.20Ex. 24.30Others

Chapter 24: The circle Exercise 24.10 solutions [Pages 21 - 22]

Ex. 24.10 | Q 1.1 | Page 21

Find the equation of the circle with:

Centre (−2, 3) and radius 4.

Ex. 24.10 | Q 1.2 | Page 21

Find the equation of the circle with:

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

Ex. 24.10 | Q 1.3 | Page 21

Find the equation of the circle with:

Centre (0, −1) and radius 1.

Ex. 24.10 | Q 1.4 | Page 21

Find the equation of the circle with:

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

Ex. 24.10 | Q 1.5 | Page 21

Find the equation of the circle with:

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

Ex. 24.10 | Q 2.1 | Page 21

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

 (x − 1)2 + y2 = 4

Ex. 24.10 | Q 2.2 | Page 21

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

(x + 5)2 + (y + 1)2 = 9

Ex. 24.10 | Q 2.3 | Page 21

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

x2 + y2 − 4x + 6y = 5

Ex. 24.10 | Q 2.4 | Page 21

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

x2 + y2 − x + 2y − 3 = 0.

Ex. 24.10 | Q 3 | Page 21

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

Ex. 24.10 | Q 4 | Page 21

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

Ex. 24.10 | Q 5 | Page 21

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

Ex. 24.10 | Q 6 | Page 21

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

Ex. 24.10 | Q 7.1 | Page 21

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

Ex. 24.10 | Q 7.2 | Page 21

Find the equation of a circle
which touches x-axis at a distance 5 from the origin and radius 6 units.

Ex. 24.10 | Q 7.3 | Page 21

Find the equation of a circle
which touches both the axes and passes through the point (2, 1).

Ex. 24.10 | Q 7.4 | Page 21

Find the equation of a circle
passing through the origin, radius 17 and ordinate of the centre is −15.

Ex. 24.10 | Q 8 | Page 21

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

Ex. 24.10 | Q 9 | Page 21

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

Ex. 24.10 | Q 10 | Page 21

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

Ex. 24.10 | Q 11 | Page 21

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.

Ex. 24.10 | Q 12 | Page 21

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

Ex. 24.10 | Q 13 | Page 21

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

Ex. 24.10 | Q 14 | Page 21

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

Ex. 24.10 | Q 15 | Page 21

If the line y = \[\sqrt{3}\] x + k touches the circle x2 + y2 = 16, then find the value of k

Ex. 24.10 | Q 16 | Page 21

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

Ex. 24.10 | Q 17 | Page 21

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

Ex. 24.10 | Q 18 | Page 21

Show that the point (xy) 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.

 

Ex. 24.10 | Q 19 | Page 22

The circle x2 + y2 − 2x − 2y + 1 = 0 is rolled along the positive direction of x-axis and makes one complete roll. Find its equation in new-position.

Ex. 24.10 | Q 20 | Page 22

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

Ex. 24.10 | Q 21 | Page 22

If the line 2x − 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.

Chapter 24: The circle Exercise 24.20 solutions [Pages 31 - 32]

Ex. 24.20 | Q 1.1 | Page 31

Find the coordinates of the centre and radius of each of the following circles:  x2 + y2 + 6x − 8y − 24 = 0

Ex. 24.20 | Q 1.2 | Page 31

Find the coordinates of the centre and radius of each of the following circles: 2x2 + 2y2 − 3x + 5y = 7

Ex. 24.20 | Q 1.3 | Page 31

Find the coordinates of the centre and radius of each of the following circles: 1/2 (x2 + y2) + x cos θ + y sin θ − 4 = 0

Ex. 24.20 | Q 1.4 | Page 31

Find the coordinates of the centre and radius of each of the following circles:  x2 y2 − ax − by = 0

Ex. 24.20 | Q 2.1 | Page 32

Find the equation of the circle passing through the points:

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

Ex. 24.20 | Q 2.3 | Page 32

Find the equation of the circle passing through the points:

 (5, −8), (−2, 9) and (2, 1)

Ex. 24.20 | Q 2.4 | Page 32

Find the equation of the circle passing through the points:

 (0, 0), (−2, 1) and (−3, 2)

Ex. 24.20 | Q 3 | Page 32

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

Ex. 24.20 | Q 4 | Page 32

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

Ex. 24.20 | Q 5 | Page 32

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

Ex. 24.20 | Q 6 | Page 32

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.

Ex. 24.20 | Q 7.1 | Page 32

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

Ex. 24.20 | Q 7.2 | Page 32

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

Ex. 24.20 | Q 7.3 | Page 32

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

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

Ex. 24.20 | Q 7.4 | Page 32

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

Ex. 24.20 | Q 8 | Page 32

Prove that the centres of the three circles x2 y2 − 4x − 6y − 12 = 0, x2 + y2 + 2x + 4y − 10 = 0 and x2 + y2 − 10x − 16y − 1 = 0 are collinear.

Ex. 24.20 | Q 9 | Page 32

Prove that the radii of the circles x2 + y2 = 1, x2 + y2 − 2x − 6y − 6 = 0 and x2 + y2 − 4x − 12y − 9 = 0 are in A.P.

Ex. 24.20 | Q 10 | Page 32

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.

Ex. 24.20 | Q 11 | Page 32

Find the equation of the circle concentric with the circle x2 + y2 − 6x + 12y + 15 = 0 and double of its area.

Ex. 24.20 | Q 12 | Page 32

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.

Ex. 24.20 | Q 13 | Page 32

Find the equation of the circle concentric with x2 + y2 − 4x − 6y − 3 = 0 and which touches the y-axis.

Ex. 24.20 | Q 14 | Page 32

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

Ex. 24.20 | Q 15 | Page 32

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.

Chapter 24: The circle Exercise 24.30 solutions [Pages 37 - 38]

Ex. 24.30 | Q 1 | Page 37

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

Ex. 24.30 | Q 2 | Page 37

Find the equation of the circle the end points of whose diameter are the centres of the circles x2 + y2 + 6x − 14y − 1 = 0 and x2 + y2 − 4x + 10y − 2 = 0.

Ex. 24.30 | Q 3 | Page 37

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.

Ex. 24.30 | Q 4 | Page 37

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

Ex. 24.30 | Q 5 | Page 37

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

Ex. 24.30 | Q 6 | Page 37

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

Ex. 24.30 | Q 7 | Page 37

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.

Ex. 24.30 | Q 8 | Page 37

The abscissae of the two points A and B are the roots of the equation x2 + 2ax − b2 = 0 and their ordinates are the roots of the equation x2 + 2px − q2 = 0. Find the equation of the circle with AB as diameter. Also, find its radius.

Ex. 24.30 | Q 9 | Page 37

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 x2 + y2 − a (x + y) = 0.

Ex. 24.30 | Q 10 | Page 37

The line 2x − y + 6 = 0 meets the circle x2 + y2 − 2y − 9 = 0 at A and B. Find the equation of the circle on AB as diameter.

Ex. 24.30 | Q 11 | Page 38

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

Ex. 24.30 | Q 12 | Page 38

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.

Chapter 24: The circle solutions [Page 38]

Q 1 | Page 38

Write the length of the intercept made by the circle x2 + y2 + 2x − 4y − 5 = 0 on y-axis.

Q 2 | Page 38

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

Q 3 | Page 38

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

Q 4 | Page 38

If the abscissae and ordinates of two points P and Q are roots of the equations x2 + 2ax − b2 = 0 and x2 + 2px − q2 = 0 respectively, then write the equation of the circle with PQ as diameter.

Q 5 | Page 38

Write the equation of the unit circle concentric with x2 + y2 − 8x + 4y − 8 = 0.

Q 6 | Page 38

If the radius of the circle x2 + y2 + ax + (1 − a) y + 5 = 0 does not exceed 5, write the number of integral values a.

Chapter 24: The circle solutions [Page 38]

Q 7 | Page 38

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

Q 8 | Page 38

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.

Q 9 | Page 38

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.

Chapter 24: The circle solutions [Pages 39 - 40]

Q 1 | Page 39

If the equation of a circle is λx2 + (2λ − 3) y2 − 4x + 6y − 1 = 0, then the coordinates of centre are

  • (4/3, −1)

  • (2/3, −1)

  • (−2/3, 1)

  • (2/3, 1)

Q 2 | Page 39

If 2x2 + λxy + 2y2 + (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

Q 3 | Page 39

The equation x2 + y2 + 2x − 4y + 5 = 0 represents

  • a point

  • a pair of straight lines

  • a circle of non-zero radius

  • none of these

Q 4 | Page 39

If the equation (4a − 3) x2 + ay2 + 6x − 2y + 2 = 0 represents a circle, then its centre is

  • (3, −1)

  • (3, 1)

  • (−3, 1)

  • none of these

Q 5 | Page 39

The radius of the circle represented by the equation 3x2 + 3y2 + λxy + 9x + (λ − 6) y + 3 = 0 is

  • \[\frac{3}{2}\]

  • \[\frac{\sqrt{17}}{2}\]

  •  2/3

  • none of these

Q 6 | Page 39

The number of integral values of λ for which the equation x2 + y2 + λx + (1 − λ) y + 5 = 0 is the equation of a circle whose radius cannot exceed 5, is

  • 14

  • 18

  • 16

  • none of these

Q 7 | Page 39

The equation of the circle passing through the point (1, 1) and having two diameters along the pair of lines x2 − y2 −2x + 4y − 3 = 0, is

  • x2 + y2 − 2x − 4y + 4 = 0

  •  x2 + y2 + 2x + 4y − 4 = 0

  • x2 + y2 − 2x + 4y + 4 = 0

  • none of these

Q 8 | Page 39

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

  • x2 + y2 − 2x − 2y − 3 = 0

  • x2 + y2 + 2x − 2y − 3 = 0

  • x2 + y2 + 2x + 2y − 3 = 0

  • none of these

Q 9 | Page 39

If the point (2, k) lies outside the circles x2 + y2 + x − 2y − 14 = 0 and x2 + y2 = 13 then k lies in the interval

  • (−3, −2) ∪ (3, 4)

  • −3, 4

  • (−∞, −3) ∪ (4, ∞)

  • (−∞, −2) ∪ (3, ∞)

Q 10 | Page 39

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

Q 11 | Page 39

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

  • x2 + y2 − 6x −6y + 9 = 0 

  • 4 (x2 + y2 − x − y) + 1 = 0

  • 4 (x2 + y2 + x + y) + 1 = 0

  • none of these

Q 12 | Page 39

If the circles x2 + y2 = 9 and x2 + y2 + 8y + c = 0 touch each other, then c is equal to

  • 15

  • -15

  • 16

  • -16

Q 13 | Page 39

If the circle x2 + y2 + 2ax + 8y + 16 = 0 touches x-axis, then the value of a is

  • ± 16

  • ±4

  • ± 8

  • ±1

Q 14 | Page 40

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

  • x2 + y2 ± 10x ± 10y + 5 = 0

  • x2 + y2 ± 10x ± 10y = 0

  • x2 + y2 ± 10x ± 10y + 25 = 0

  • x2 + y2 ± 10x ± 10y + 51 = 0

Q 15 | Page 40

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

  • x2 + y2 − 12x − 16y = 0

  • x2 + y2 + 12x + 16y = 0

  • x2 + y2 + 6x + 8y = 0

  • x2 + y2 − 6x − 8y = 0

Q 16 | Page 40

The equation of the circle concentric with x2 + y2 − 3x + 4y − c = 0 and passing through (−1, −2) is

  •  x2 + y2 − 3x + 4y − 1 = 0

  • x2 + y2 − 3x + 4y = 0

  • x2 + y2 − 3x + 4y + 2 = 0

  • none of these

Q 17 | Page 40

The circle x2 + y2 + 2gx + 2fy + c = 0 does not intersect x-axis, if

  • g2 < c

  • g2 > c

  • g2 > 2c

  • none of these

Q 18 | Page 40

The area of an equilateral triangle inscribed in the circle x2 + y2 − 6x − 8y − 25 = 0 is

  • \[\frac{225\sqrt{3}}{6}\]

  •  25π

  • 50π − 100

  • none of these

Q 19 | Page 40

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 x2 + y2 − 2cx − 2cy + c2 = 0, where c is equal to

  • 4

  • 2

  • 3

  • 6

Q 20 | Page 40

If the circles x2 + y2 = a and x2 + y2 − 6x − 8y + 9 = 0, touch externally, then a =

  • 1

  • -1

  • 21

  • 16

Q 21 | Page 40

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

Q 22 | Page 40

If (−3, 2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0 which is concentric with the circle x2 + y2 + 6x + 8y − 5 = 0, then c =

  • 11

  • -11

  • 24

  • none of these

Q 23 | Page 40

Equation of the diameter of the circle x2 + y2 − 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.

Q 24 | Page 40

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

  •  x2 + y2 + ax + by = 0

  • x2 + y2 − ax − by = 0

  • x2 + y2 + bx + ay = 0

  • none of these

Q 25 | Page 40

If the circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 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

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

Chapter 24: The circle

Ex. 24.10Ex. 24.20Ex. 24.30Others

RD Sharma Mathematics Class 11

Mathematics 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.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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.

Using RD Sharma Class 11 solutions The circle exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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