KC Sinha Mathematics Solution Class 11 Chapter 22 Conic Section: Circle Exercise 22.2

 Exercise 22.1 

Page no 22.30

Type 1

Question 1

Find the center and radius of the following circles :

(i) $x^{2}+y^{2}-8 x-12 y-48=0$

(ii) $x^{2}+y^{2}-a x-b y=0$

(iii) $3 x^{2}+3 y^{2}+12 x-18 y-11=0$

(iv) $x^{2}+y^{2}-2 x+4 y=8$

(v) $\frac{1}{2}\left(x^{2}+y^{2}\right)+x \cos \theta+y \sin \theta-4=0$

Question 2

Prove that the centers of the circles $x^{2}+y^{2}=1$, $x^{2}+y^{2}+6 x-2 y-1=0$ and $x^{2}+y^{2}-12 x+4 y=1$ are collinear.

Question 3

Prove that the centers of the three circles

$\begin{gathered}x^{2}+y^{2}-4 x-6 y-14=0 \\x^{2}+y^{2}+2 x+4 y-5=0, \text { and } \\x^{2}+y^{2}-10 x-16 y+7=0\end{gathered}$

are collinear.

Question 4

Prove that the radii of the circles $x^{2}+y^{2}=1, x^{2}+y^{2}-2 x-6 y=6$ and $x^{2}+y^{2}-4 x-12 y=9$ are in arithmetic progression.

Question 5

Prove that the radii of the circles $x^{2}+y^{2}=4,4 x^{2}+4 y^{2}-8 x-24 y+15=0$ and $x^{2}+y^{2}-4 y-5=0$ arc in arithmetic progression.

Type 2

Question 6

Find the equation of the circles passing through the following three points :

(i) $(0,0),(5,0)$ and $(3,3)$

(ii) $(1,0),(0,1)$ and $(-1,0)$

(iii) $(1,-2),(5,4)$ and $(10,5)$

(iv) $(1,2),(3,-4)$ and $(5,-6)$

Question 7

Fin the equation of the circle circumscribing the triangle formed by the line . 

$x+y=6, \quad 2 x+y=4$ and $x+2 y=5$

Page no 22.31

Question 8

(i) Find the equation of the circle which is concentric with the $x^{2}+y^{2}-4 x+6 y-3=0$ and the double of its area.

(ii) Find the equation of a circle concentric with the circle $2 x^{2}+2 y^{2}-6 x+8 y+1=0$ and of double its area.

Question 9

Find the equation of the circle concentric with the circle $x^{2}+y^{2}+4 x-8 y-6=0$ and having radius double of its radius.

Question 10

Find the equation of the circle concentric with the circle $x^{2}+y^{2}-4 x-6 y-9=0$ and passing through the point $(-4,-5)$.

Question 11

Find the equation of the circle passing through the points $(1,-1)$ and center at the intersection of the lines $x-y=4$ and $2 x+3 y=-7$.

Question 12

The line $5 x-y=3$ is tangent to a circle at the point $(2,7)$ and its center is on the line $x+2 y=19$. Find the equation of the circle.

Question 13

 The line $4 x-3 y=-12$ is tangent at the point $(-3,0)$ and the line $3 x+4 y=16$ is tangent at the point $(4,1)$ to a circle. Find the equation of the circle.

Question 14

Find the equation of the circle circumscribing the quadrilateral formed by the straight  lines $x-y=0,3 x+2 y=5, x-y=10$ and $2 x+3 y=0$

Question 15

(i) Find the equation of the circle passing through the points $(0,-1)$ and $(2,0)$ and whose center lies on the line $3 x+y=5$.

(ii) Find the equation of the circle passing through the points $(2,-3)$ and $(3,-2)$ and whose center lies on the line $2 x-3 y=8$

Page no 22.30

Type 3

Question 16

Determine whether the following equations represent a circle or not :

(i) $3 x^{2}-3 y^{2}+4 x-6 y+10=0$

(ii) $5 x^{2}+5 y^{2}+2 x y+4 x-y+2=0$

(iii) $5 x^{2}+5 y^{2}+4 x-8 y-16=0$

(iv) $x^{2}+y^{2}+6 x-8 y+50=0$

Question 17

Determine whether the following equations represent a circle, a point or no circle

(i) $x^{2}+y^{2}+x-y=0$

(ii) $x^{2}+y^{2}-6 x-8 y+25=0$

(iii) $x^{2}+y^{2}+2 x+10 y+26=0$

(iv) $2 x^{2}+2 y^{2}-24 x+8 y+120=0$