Question

There is a rectangle drawn such that their diagonals meet at the origin and their sides are parallel to the coordinate axes. The length of the sides parallel to y-axis is equal to the length of the conjugate axis of the hyperbola and the length of other two sides is equal to the distance between focii of the hyperbola. Then the area bounded by the hyperbola and the rectangle, is

• A. $\frac{45}{8}-6\ln 2$
• B. $\frac{45}{2}-24\ln 2$
• C. $\frac{45}{8}+6\ln 2$
• D. $\frac{45}{2}+24\ln 2$

Answer 1

The correct option is B $\frac{45}{2}-24\ln 2$

$\frac{x^2}{16}-\frac{y^2}{9}=1$
$a=4,b=3$
$\therefore e=\sqrt{1+\frac{b^2}{a^2}}=\frac{5}{4}$
Focii are $(5,0)$ and $(-5,0)$

For $x=5,y=3\sqrt{\frac{25}{16}-1}=\frac{9}{4}$
So, point of intersection is $(5,\frac{9}{4})$


Required area $=4\times \left[\frac{3}{4}{\int }_4^5\sqrt{x^2-16}dx\right]$
$=3{[\frac{x}{2}\sqrt{x^2-16}-8\ln |x+\sqrt{x^2-16}|]}_4^5$
$=3[\frac{15}{2}-8\ln 2]$
$=\frac{45}{2}-24\ln 2$