Question

X is a point out side of a circle whose centre 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

The correct option is A $\frac{3a}{4}$

Let the $AX$ be the shortest distance from circle and $BX$ be the tangent

Given $AX=\frac{a}{2},BX=a$

Let $r$ be the radius of the circle

As tangent will be perpendicular to the radius

So $\angle XBO={90}^{\circ }$

In $△XBO,\angle XBO={90}^{\circ }$

So $O{X}^2=O{B}^2+B{X}^2$

$⟹(\frac{a}{2}+r{)}^2=r^2+a^2$

$⟹\frac{a^2}{4}+r^2+2(\frac{a}{2})\left(r\right)=r^2+a^2$ $(\because (a+b{)}^2=a^2+b^2+2ab)$

$⟹ar=a^2-\frac{a^2}{4}$

$⟹ar=\frac{3a^2}{4}$

$⟹r=\frac{3a}{4}$

So the radius of the circle is $\frac{3a}{4}$

Hence the answer is $\text{Opt : A}$