Question

For real number a, b (a > b > 0), let
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)
and 
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)
Then the value of (a – b)² is equal to _____.

Answer 1

Correct Answer: 12

The correct answer is 12
\(x²+y² ≤ a²\) is interior of circle and\(\frac{x²}{a²} + \frac{y²}{b²} ≥ 1\) is exterior of ellipse

Fig. 

∴ Area = πa² - πab = 30π .... (1)
Similarly \(x²+y² ≥ b²\) and \(\frac{x²}{a²} + \frac{y²}{b²} ≤ 1\) gives
 πab - πb² = 30π .... (2)
Comparing (1) and (2) , \(\frac{a}{b}=\frac{5}{3}\) and \(⇒ a=\frac{5b}{3}\)
\(⇒ π.\frac{25b²}{9} - π . \frac{5b²}{3} = 30π\)
\(⇒ ( \frac{25}{9} - \frac{5}{3} )b² = 30\)
\(⇒ \frac{10}{9} b² = 30 ⇒ b² = 27\)
and \(a² = \frac{25}{9} . 27 = 75\)
\(= (a-b)²\)
\(= (5\sqrt3 - 3\sqrt3)² \)
\(= 3.4\)
= 12