Question

A variable straight line $AB$ divides the circumference of the circle $x^2+y^2=25$ in the ratio $1:2.$ If a tangent $CD$ is drawn to the smaller arc parallel to $AB,$ such that $ABCD$ is a rectangle $($as shown in the figure$),$ then locus of $C$ and $D$ is

• A. $x^2+y^2=\frac{175}{4}$
• B. $x^2+y^2=36$
• C. $x^2+y^2=40$
• D. $x^2+y^2=20$

Answer 1

The correct option is A $x^2+y^2=\frac{175}{4}$


$\angle AOB=\frac{1}{(1+2)}\times {360}^{\circ }={120}^{\circ }$
$\therefore \angle AON=\frac{1}{2}\angle AOB={60}^{\circ }$
From figure,
$AM=DN=5\sin {60}^{\circ }=\frac{5\sqrt{3}}{2}$

Now in $△OND,$
${\left(OD\right)}^2={\left(DN\right)}^2+{\left(ON\right)}^2$
$\Rightarrow (OD{)}^2=\frac{75}{4}+25$
$\Rightarrow (OD{)}^2=\frac{175}{4}$
So, locus of $C$ and $D$ is $x^2+y^2=\frac{175}{4}$