Question

Let the tangents from $P$ are drawn to a circle with centre $C$ such that $PC=2$ units. If equation of the tangents are $x^2=3y^2,$ then radius of the circle can be

• A. $\sqrt{2}$ unit
• B. $\sqrt{3}$ unit
• C. $2$ unit
• D. $1$ unit

Answer 1

Answer: (B), (D)

$1$ unit

$x^2=3y^2\Rightarrow x=\sqrt{3}y;x=-\sqrt{3}y$
So $P\equiv (0,0)$
Now the centre of the required circle will lie on the angle bisector of the lines i.e., $x=0;y=0$
and it is given that $PC=2$ units

Now from above figure in $△P{P}_1{C}_1$
$\angle C_{1}P{P}_1=30°$
$\Rightarrow r=C_1{P}_1=P{C}_1\sin 30°=1$ unit

Now in $△P{P}_2{C}_2$
$\angle C_{2}P{P}_2=60°$
$\Rightarrow r=C_2{P}_2=P{C}_2\sin 60°=\sqrt{3}$ unit