Question

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a,b>0)$ passes through the point of intersection of the lines $7x+13y-87=0$ & $5x-8y+7=0$ and the latus rectum is $\frac{32\sqrt{2}}{5}$. The values of $a$ and $b$ are:

• A. $a=\frac{5}{\sqrt{2}},b=3$.
• B. $a=\frac{5}{\sqrt{2}},b=4$.
• C. $a=\frac{7}{\sqrt{2}},b=3$.
• D. None of these

Answer 1

The correct option is B $a=\frac{5}{\sqrt{2}},b=4$.

Point of intersection of lines
$7x+13y-87=0$ & $5x-8y+7=0$ is $(5,4)$.
Also this point lies on the given hyperbola
$\therefore \frac{25}{a^2}-\frac{16}{b^2}=1$ ......(1)
Also latus rectum $LR=\frac{2b^2}{a}=\frac{32\sqrt{2}}{5}$
$\Rightarrow b^2=\frac{16\sqrt{2}a}{5}$ .....(ii)
From (i) & (ii) $a^2=\frac{25}{2},b^2=16$.