CAT 2015 | Question: 72

Question

$\text{PQRS}$ is a square. $\text{SR}$ is a tangent (at point $\text{S})$ to the circle with centre $\text{O}$ and $\text{TR = OS}$. Then the ratio of area of the circle to the area of the square is 

• A. $\pi /3$
• B. $11/7$
• C. $3 /\pi$
• D. $7/11$

Answer 1

Given that the figure.

Let the radius of a circle be $r$ cm.

DIAGRAM

The triangle $ \triangle \; \text{OSR}$ is right-angle triangle. so we can apply the Pythagorean theorem.

$\text{(Hypotenuse)}^{2} = \text{(Perpendicular)}^{2} + \text{(Base)}^{2}$

$\Rightarrow \text{(OR)}^{2} = \text{(SR)}^{2} + \text{(SO)}^{2}$

$\Rightarrow \text{(2r)}^{2} = \text{(SR)}^{2} + r^{2}$

$\Rightarrow 4r^{2} – r^{2} = \text{(SR)}^{2} $

$\Rightarrow \text{(SR)}^{2}  = 3r^{2} $

$\Rightarrow  \boxed{\text{SR} = \sqrt{3} \; r \; cm }$

Now, we can calculate the area of circle and square.

• Area of circle $ = \pi \; r^{2} \;  cm^{2}$
• Area of square $ = 3r^{2} \;  cm^{2}$

$\therefore$ The required ratio $ = \pi \; r^{2} : 3r^{2} =  \pi : 3$

Correct Answer $: \text{A}$