question_answer

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)         \[a\]

Correct Answer: A

Solution :

 Here,    \[PX=a\] \[OP=r\] and        \[AX=\frac{a}{2}\] Since PX is a tangent                 \[\therefore \]  \[\angle OPX={{90}^{o}}\] \[\therefore \]  \[O{{P}^{2}}+P{{X}^{2}}=O{{X}^{2}}\] or            \[{{r}^{2}}+{{a}^{2}}={{\left( r+\frac{a}{2} \right)}^{2}}\] or            \[{{r}^{2}}+{{a}^{2}}={{r}^{2}}+\frac{{{a}^{2}}}{4}+ar\] or            \[ar=\frac{3{{a}^{2}}}{4}\] or            \[r=\frac{3a}{4}\]