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 $\theta$.
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