Question

The acute angle between the common tangents of two circles $x^2+y^2=25$ and $(x-10{)}^2+y^2=100$ is

• A. $60°$
• B. $75°$
• C. $15°$
• D. $30°$

Answer 1

The correct option is A $60°$

Let the common tangents meet at $(x,y)$
$\because$ Centers of both the circle lie on the x-axis, so y-coordinate of the point will be $0$.

From the above figure, $\frac{O{C}_1}{O{C}_2}=\frac{10}{5}=\frac{2}{1}$
$C_1{C}_2=O{C}_1-O{C}_2\Rightarrow O{C}_1=10+O{C}_2$

$\frac{10+O{C}_2}{O{C}_2}=\frac{2}{1}\Rightarrow O{C}_2=10$
So, the coordinates of the point is $(-10,0)$
Now, equation of the tangents drawn from the point $(-10,0)$ is
$y=mx+a\sqrt{1+m^2}$
$0=-10m+5\sqrt{1+m^2}$
$\Rightarrow m=\pm \frac{1}{\sqrt{3}}$
$\therefore \theta =60°$