Position of a Point With Respect to Circle - Practice Questions & MCQ

Position of a Point With Respect to Circle

Let $S$ be a circle and $P$ be any point in the plane. Then
$S: x^2+y^2+2 g x+2 f y+c=0$
Centre of the circle, $\mathrm{C}(-\mathrm{g},-\mathrm{f})$

To check if the point $P\left(x_1, y_1\right)$ lies outside, on or inside the circle $S$
- If $\mathrm{CP}^2-\mathrm{r}^2>0$, then CP is greater than r , which means P lies outside the circle
- If $C P^2-r^2<0$, then $C P$ is lesser than $r$, which means $P$ lies inside the circle
- If $\mathrm{CP}^2-\mathrm{r}^2=0$, then CP is equal to r , which means P lies on the circle

Let us find the equation for $\mathrm{CP}^2-\mathrm{r}^2$

$
\begin{aligned}
& \left(x_1-(-g)\right)^2+\left(y_1-(-f)\right)^2-\left(g^2+f^2-c\right) \\
& x_1^2+g^2+2 g x_1+y_1^2+f^2+2 f y_1-\left(g^2+f^2-c\right) \\
& x_1^2+y_1^2+2 g x_1+2 f y_1+c
\end{aligned}
$

This expression can also be obtained by substituing the coordinates of point P in the equation of the circle, and we call this expression $\mathrm{S}_1$
So, if
(a) P lies outside the circle $\Leftrightarrow \mathrm{S}_1>0$
(b) P lies on the circle (on the circumference) $\Leftrightarrow \mathrm{S}_1=0$
(c) P lies inside the circle $\Leftrightarrow \mathrm{S}_1<0$

Greatest and Least Distance of a Point from a Circle
$S_1$ be a circle and $P$ be any point in the plane.

$
\begin{aligned}
& S_1: x^2+y^2+2 g x+2 f y+c=0 \\
& P=\left(x_1, y_1\right)
\end{aligned}
$

The centre of circle is $\mathrm{C}(-\mathrm{g},-\mathrm{f})$ and radius r is $\sqrt{g^2+f^2-c}$

(a) If P lies inside of the circle

The minimum distance of P from the circle = PA = AC - PC =  r - PC  

The maximum distance of P from the circle = PB = BC + PC = r + PC 

(b) If P lies outside of the circle

The minimum distance of P from the circle = PA = CP - AC = CP - r

The maximum distance of P from the circle = PB = BC + PC = r + PC