Question

$X$ is a point outside of a circle whose center is $O$. From $X$, a tangent whose length is $a$ is drawn to the circle and the shortest distance from $X$ to the circle is $\frac{a}{2}$. Find the radius of the circle.

• A. $\frac{3a}{4}$
• B. $\frac{3a}{2}$
• C. $\frac{1}{2}a$
• D. $a$

Answer 1

Answer: (A)

$\frac{3a}{4}$

The given circle has centre at $O$, radius $OP=OA=r,PX=a$ is the tangent to the circle at $P$ and $AX=\frac{a}{2}$.(AX is the shortest length )

$\therefore \angle OPX={90}^o$ i.e. $\Delta OPX$ is a right triangle , the hypotenuse being $OX$.

Now $AX=\frac{a}{2},OX=OA+AX=r+\frac{a}{2},PX=a$ and $OP=r$.

$\therefore {OP}^2+{PX}^2={OX}^2$

$\Rightarrow r^2+a^2={(r+\frac{a}{2})}^2$

$\Rightarrow ar=\frac{3a^2}{4}=r=\frac{3a}{4}$