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Chapters
2: Relations
3: Functions
4: Measurement of Angles
5: Trigonometric Functions
6: Graphs of Trigonometric Functions
7: Values of Trigonometric function at sum or difference of angles
8: Transformation formulae
9: Values of Trigonometric function at multiples and submultiples of an angle
10: Sine and cosine formulae and their applications
11: Trigonometric equations
12: Mathematical Induction
13: Complex Numbers
14: Quadratic Equations
15: Linear Inequations
16: Permutations
17: Combinations
18: Binomial Theorem
19: Arithmetic Progression
20: Geometric Progression
21: Some special series
22: Brief review of cartesian system of rectangular co-ordinates
23: The straight lines
▶ 24: The circle
25: Parabola
26: Ellipse
27: Hyperbola
28: Introduction to three dimensional coordinate geometry
29: Limits
30: Derivatives
31: Mathematical reasoning
32: Statistics
33: Probability
![RD Sharma solutions for Mathematics [English] Class 11 chapter 24 - The circle - Shaalaa.com](/images/9788193663004-mathematics-english-class-11_6:972cafaba17f4949992ada196fa0f041.jpg "RD Sharma solutions for Mathematics [English] Class 11 chapter 24 - The circle")
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Solutions for Chapter 24: The circle
Below listed, you can find solutions for Chapter 24 of CBSE, Karnataka Board PUC RD Sharma for Mathematics [English] Class 11.
RD Sharma solutions for Mathematics [English] Class 11 24 The circle Exercise 24.1 [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 + y2 = 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:
x2 + y2 − 4x + 6y = 5
Find the centre and radius of each of the following circles:
x2 + y2 − x + 2y − 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 + 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.
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 2x + y = 6 and 3x + 2y = 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 5x + 12y − 1 = 0.
Find the equation of the circle which touches the axes and whose centre lies on x − 2y = 3.
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.
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 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.
If the line y = \[\sqrt{3}\] x + 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 the intersection of the lines 3x + y = 14 and 2x + 5y = 18.
If the lines 3x − 4y + 4 = 0 and 6x − 8y − 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 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.
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.
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.
RD Sharma solutions for Mathematics [English] Class 11 24 The circle Exercise 24.2 [Pages 31 - 32]
Find the coordinates of the centre and radius of each of the following circles: x2 + y2 + 6x − 8y − 24 = 0
Find the coordinates of the centre and radius of each of the following circles: 2x2 + 2y2 − 3x + 5y = 7
Find the coordinates of the centre and radius of the following circle:
1/2 (x2 + y2) + x cos θ + y sin θ − 4 = 0
Find the coordinates of the centre and radius of each of the following circles: x2 + y2 − 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 2x − 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 − 4y = 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 2x + y − 3 = 0, x + y − 1 = 0 and 3x + 2y − 5 = 0
Find the equation of the circle which circumscribes the triangle formed by the lines
x + y = 2, 3x − 4y = 6 and x − y = 0.
Find the equation of the circle which circumscribes the triangle formed by the lines y = x + 2, 3y = 4x and 2y = 3x.
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.
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.
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 x2 + y2 − 6x + 12y + 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 x2 + y2 − 4x − 6y − 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 − 4x + 3 = 0.
RD Sharma solutions for Mathematics [English] Class 11 24 The circle Exercise 24.3 [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 x2 + y2 + 6x − 14y − 1 = 0 and x2 + y2 − 4x + 10y − 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 − 3y = 4, 3x + y = 22, x − 3y = 14 and 3x + y = 62.
Find the equation of the circle passing through the origin and the points where the line 3x + 4y = 12 meets the axes of coordinates.
Find the equation of the circle which passes through the origin and cuts off intercepts aand 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 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.
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.
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.
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.
RD Sharma solutions for Mathematics [English] Class 11 24 The circle Exercise 24.4 [Page 38]
Write the length of the intercept made by the circle x2 + y2 + 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 x2 + 2ax − b2 = 0 and x2 + 2px − q2 = 0 respectively, then write the equation of the circle with PQ as diameter.
Write the equation of the unit circle concentric with x2 + y2 − 8x + 4y − 8 = 0.
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.
RD Sharma solutions for Mathematics [English] Class 11 24 The circle Exercise 24.5 [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.
RD Sharma solutions for Mathematics [English] Class 11 24 The circle Exercise 24.6 [Pages 39 - 40]
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)
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
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
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
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
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
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
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
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, ∞)
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
x2 + y2 − 6x −6y + 9 = 0
4 (x2 + y2 − x − y) + 1 = 0
4 (x2 + y2 + x + y) + 1 = 0
none of these
If the circles x2 + y2 = 9 and x2 + y2 + 8y + c = 0 touch each other, then c is equal to
15
-15
16
-16
If the circle x2 + y2 + 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
x2 + y2 ± 10x ± 10y + 5 = 0
x2 + y2 ± 10x ± 10y = 0
x2 + y2 ± 10x ± 10y + 25 = 0
x2 + y2 ± 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
x2 + y2 − 12x − 16y = 0
x2 + y2 + 12x + 16y = 0
x2 + y2 + 6x + 8y = 0
x2 + y2 − 6x − 8y = 0
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
The circle x2 + y2 + 2gx + 2fy + c = 0 does not intersect x-axis, if
g2 < c
g2 > c
g2 > 2c
none of these
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
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
If the circles x2 + y2 = a and x2 + y2 − 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 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
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.
∴
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
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
\[\frac{1}{a} + \frac{1}{b} = \frac{2}{c}\]
Solutions for 24: The circle
RD Sharma solutions for Mathematics [English] Class 11 chapter 24 - The circle
Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Mathematics [English] Class 11 CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. RD Sharma solutions for Mathematics Mathematics [English] Class 11 CBSE, Karnataka Board PUC 24 (The circle) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
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Concepts covered in Mathematics [English] Class 11 chapter 24 The circle are Sections of a Cone, Introduction of Parabola, Standard Equations of Parabola, Latus Rectum, Introduction of Ellipse, Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus from the Centre of the Ellipse, Special Cases of an Ellipse, Concept of Circle, Standard Equations of an Ellipse, Latus Rectum, Introduction of Hyperbola, Eccentricity, Standard Equation of Hyperbola, Latus Rectum, Standard Equation of a Circle, Eccentricity.
Using RD Sharma Mathematics [English] Class 11 solutions The circle exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in RD Sharma Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Mathematics [English] Class 11 students prefer RD Sharma Textbook Solutions to score more in exams.
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