Circle - Practice Questions & MCQ

Circle

Definition

A circle is the locus of a moving point such that its distance from a fixed point is constant.

The fixed point is called the centre (O) of the circle and constant distance is called its radius (r)

Equation of circle

Centre-Radius Form

The equation of a circle with centre at $C(h, k)$ and radius $r$ is $(x-h)^2+(y-k)^2=r^2$

Let $P(x, y)$ be any point on the circle. Then, by definition, $|C P|=r$.
Using the distance formula, we have

$
\sqrt{(\mathrm{x}-\mathrm{h})^2+(\mathrm{y}-\mathrm{k})^2}=\mathrm{r}
$

i.e.

$
(\mathrm{x}-\mathrm{h})^2+(\mathrm{y}-\mathrm{k})^2=\mathrm{r}^2
$
If the centre of the circle is the origin or $(0,0)$ then equation of the circle becomes

$
\begin{aligned}
& (x-0)^2+(y-0)^2=r^2 \\
& \text { i.e. } x^2+y^2=r^2
\end{aligned}
$
General Form:
The equation of a circle with centre at $(\mathrm{h}, \mathrm{k})$ and radius r is

$
\begin{aligned}
& \Rightarrow(x-h)^2+(y-k)^2=r^2 \\
& \Rightarrow x^2+y^2-2 h x-2 k y+h^2+k^2-r^2=0
\end{aligned}
$
Which is of the form :

$
\mathbf{x}^2+\mathrm{y}^2+2 \mathrm{gx}+2 \mathbf{y}+\mathbf{c}=\mathbf{0}
$
This is known as the general equation of the circle.
To get radius and centre if only the equation of the circle (ii) is given:
Compare eq (i) and eq (ii)

$
\mathrm{h}=-\mathrm{g}, \mathrm{k}=-\mathrm{h} \text { and } \mathrm{c}=\mathrm{h}^2+\mathrm{k}^2-\mathrm{r}^2
$
Coordinates of the centre $(-g,-\mathrm{f})$
Radius $=\sqrt{g^2+f^2-c}$

Nature of the Circle
For the standard equation of a circle $\mathrm{x}^2+\mathrm{y}^2+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c}=0$ whose radius is given as $\sqrt{g^2+f^2-c}$
Now the following cases arise
1. If $g^2+f^2-c>0$, then the radius of the circle will be real. Hence, the circle is a real circle.
2. If $g^2+f^2-c=0$, then the radius of the circle will be real $(=0)$. Hence, the circle is a Point circle because the radius is 0 .
3. If $g^2+f^2-c<0$, then the radius of the circle will be imaginary. Hence, the circle is an imaginary circle.