Explain parametric equation of a circle. - Practice Questions & MCQ

Parametric Form

To represent any point on a curve in terms of a single variable (parameter), we use parametric form of that curve.
1. Parametric Form for $x^2+y^2=r^2$
$P(x, y)$ is a point on the circle $x^2+y^2=r^2$ with centre $O(0,0)$. And $O P$ makes an angle $\theta$ with the positive direction of $X$-axis, then $x=r \cdot \cos \theta, y=r \cdot \sin \theta$ called the parametric equation of the circle.
Here as $\theta$ varies, the point on the circle also changes, and thus $\theta$ is called the parameter. Here $0 \leq \theta<2 \pi$.
So any arbitrary point on this circle can be assumed as (r. $\cos \theta, r \cdot \sin \theta)$

2. Parametric Form for $(x-h)^2+(y-k)^2=r^2$

Centre of the circle here is $(h, k)$
Parametric point on it is $(\mathrm{h}+\mathrm{r} \cdot \cos \theta, \mathrm{k}+\mathrm{r} \cdot \sin \theta)$

Concentric Circles

Two circles having common centre $\mathrm{C}(\mathrm{h}, \mathrm{k})$ but different radii $r_1$ and $\mathrm{r}_2$ are called concentric circles

$
\begin{aligned}
& S_1=(x-h)^2+(y-k)^2=r_1^2 \\
& S_2=(x-h)^2+(y-k)^2=r_2^2 \\
& r_1 \neq r_2 \\
& \mathrm{~S}_1 \text { and } \mathrm{S}_2 \text { are concentric circle }
\end{aligned}
$