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

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