Intersection of Two Circle - Practice Questions & MCQ

Intersection of Two Circle

There are different cases of intersection of two circlre
Let the two circles be $\left(x-x_1\right)^2+\left(y-y_1\right)^2=r_1^2$ and $\left(x-x_2\right)^2+\left(y-y_2\right)^2=r_2^2$ where centres are $C_1\left(x_1, y_1\right)$ and $C_2\left(x_2, y_2\right)$ and radii are $r_1$ and $r_2$, respectively.

CASE 1
When two circles do not intersect

$
\mathrm{C}_1 \mathrm{C}_2>\mathrm{r}_1+\mathrm{r}_2
$

i.e., the distance between the centres is greater than the sum of radii, then two circles neither intersect nor touch each other.

In this case, four common tangents can be drawn to two circles, in which two are direct common tangents and the other two are transverse common tangents.

Here, point $D$ divides $C_1$ and $C_2$ internally in the ratio $r_1 \cdot I_2$ and point $P$ divides $C_1$ and $C_2$ externally in the ratio $r_1 \cdot I_2$
Then the co-ordinates of P and D are

$
\begin{aligned}
& D \equiv\left(\frac{r_1 x_2+r_2 x_1}{r_1+r_2}, \frac{r_1 y_2+r_2 y_1}{r_1+r_2}\right)=(\alpha, \beta) \\
& P \equiv\left(\frac{r_1 x_2-r_2 x_1}{r_1-r_2}, \frac{r_1 y_2-r_2 y_1}{r_1-r_2}\right)=(\gamma, \delta)
\end{aligned}
$
The transverse common tangents will pass through the point $D(\alpha, \beta)$. The equation of transverse common tangents is $(y-\beta)=m_1(x-\alpha)$
The direct common tangents will pass through the point $\mathrm{P}\left(\mathrm{y}, \bar{\delta}\right.$ ) so equation of tangents will be $(\mathrm{y}-\bar{\delta})=\mathrm{m}_2(\mathrm{x}-\mathrm{y})$.
Now values of $m_1$ and $m_2$ can be obtained from the length of the perpendicular from the centre $C_1$ or $C_2$ on the tangent is equal to $r_1$ or $r_2$. Put two values of $m_1$ and $m_2$ on the common tangent equations, then we get the required results.

CASE 2
When two circles touch each other externally

$
\mathrm{C}_1 \mathrm{C}_2=\mathrm{r}_1+\mathrm{r}_2
$

i.e, the distance between the centres is equal to the sum of radii, then two circles touch externally.

In this case, two direct common tangents are real and distinct while the transverse tangents are coincident.

Direct common tangent can be found as done in case 1

For Transverse common tangent
The equation of tangent at point P is $S_1-S_2=0$, where $S_1=0$ and $S_2=0$ are equations of the circles

CASE 3
When two circles intersect at 2 distinct points

$
\left|\mathrm{r}_1-\mathrm{r}_2\right|<\mathrm{C}_1 \mathrm{C}_2<\mathrm{r}_1+\mathrm{r}_2
$

Thus two common tangents can be drawn

Direct common tangent can be found as done in case 1

CASE 4
When two circles touch each other internally

$
\mathrm{C}_1 \mathrm{C}_2=\left|\mathrm{r}_1-\mathrm{r}_2\right|
$
Thus, only one tangent can be drawn. Equation of the common transverse tangent is

$
S_1-S_2=0
$
 

CASE-5
When one circle lies inside the other one

$
\mathrm{C}_1 \mathrm{C}_2<\left|\mathrm{r}_1-\mathrm{r}_2\right|
$
Thus, no common tangent can be drawn