KC Sinha Mathematics Solution Class 11 Chapter 22 Conic Section: Circle Exercise 22.4

 Exercise 22.4

Page no 22.42

Type 1 

Question 1

Find the parametric equations of the following circles :

(i) $x^{2}+y^{2}=9$

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

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

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

Question 2

Find the cartesian equations of the following curves whose parametric equations are :

(i) $x=5 \cos \theta, y=5 \sin \theta$

(ii) $x=a+c \cos \alpha, y=b+c \sin \alpha$

(iii) $x=3 \cos \alpha \quad y=3 \sin \alpha$

(iv) $x=1+3 \cos \theta, y=2-3 \sin \theta$

(v) $x=\cos \theta+\sin \theta, y=\sin \theta-\cos \theta$

(vi) $x=\frac{20 t}{4+t^{2}}, y=\frac{5\left(4-t^{2}\right)}{4+t^{2}}$

Question 3

If curve is a circle, find its center and radius.

Prove that : 

$x \cos \theta+y \sin \theta=a \text { and } x \sin \theta-y \cos \theta=b$

are the parametric equations of a circle for all $\theta$ satisfying $0 \leq \theta<2 \pi$.

Page no 22.43

Question 4

Show that

$x=a \cos \theta-b \sin \theta \text { and } y=a \sin \theta+b \cos \theta$

represent a circle where $\theta$ is the parameter.

Question 5

Show that the point $(x, y)$, where

$x=5 \cos \theta, y=-3+5 \sin \theta$

lies on a circle for all values of $\theta$.

Question 6

Show that the point $\left(x_{1}, y\right)$. where

$x=a+r \cos \theta, y=b+r \sin \theta \text {, }$

lies on a circle for all values of $\theta$.