Line and Circle - Practice Questions & MCQ

Line and Circle

$S$ is a circle with center $O$ and radius $r$, and $L$ is a straight line in the plane of the circle.
Equation of circle $S: x^2+y^2=a^2$
Equation of line $L: y=m x+c$
To find their point(s) of intersection, we can solve these equations simultaneously

$
\begin{aligned}
& x^2+(m x+c)^2=a^2 \\
& \left(1+m^2\right) x^2+2 m c x+c^2-a^2=0
\end{aligned}
$

Case (1)

If line L intersects the circle S in two distinct points, then Equation (iii) will have two real and distinct roots 

So, Discriminant of equation (iii) > 0

In this case the line is a secant to the circle, i.e., it represents the equation of chord to the circle.

Case (2)

If the line $L$ touches the circle, then Equation (iii) will have two equal real roots
So, Discriminant of equation (iii) $=0$

$
\begin{aligned}
& \mathrm{B}^2-4 \mathrm{AC}=0 \\
& 4 \mathrm{~m}^2 \mathrm{c}^2-4\left(1+\mathrm{m}^2\right)\left(\mathrm{c}^2-\mathrm{a}^2\right)=0 \\
& \mathrm{a}^2=\frac{\mathrm{c}^2}{1+\mathrm{m}^2} \\
& c^2=a^2\left(1+m^2\right)
\end{aligned}
$

In this case the line is a tangent to the circle

This is also the condition of tangency to the circle.

Case (3)

If the line L and the circle S have no common points, then Equation (iii) will have imaginary roots

So, Discriminant of equation (iii) < 0

Alternate method to check position of line with respect to the circle

We can find the distance OM between the line and the centre of the circle, and compare it with radius of the circle

If $\mathrm{OM}<\mathrm{r}$, then line is a chord (secant) to the circle
If $\mathrm{OM}=r$, then line is a tangent to the circle
If $\mathrm{OM}>\mathrm{r}$, then line does not intersect the circle at any point