Question

Focus of hyperbola is $(\pm 3,0)$ and equation of tangent is $2x+y-4=0$, find the equation of hyperbola.

• A. $4x^2-5y^2=20$
• B. $5x^2-4y^2=20$
• C. $4x^2-5y^2=1$
• D. $5x^2-4y^2=1$

Answer 1

The correct option is A $4x^2-5y^2=20$

Given, $(\pm ae,0)=(\pm 3,0)$
$\Rightarrow ae=3$
$\Rightarrow a^2{e}^2=9$
$\Rightarrow b^2+a^2=9$ ......$\left(i\right)$
$\because 2x+y-4=0$
$\Rightarrow y=-2x+4$
Is the tangent to the hyperbola.
$\therefore (4{)}^2=a^2(-2{)}^2-b^2$
$\Rightarrow 4a^2-b^2=16$ .......$\left(ii\right)$
On solving Eqs. $\left(i\right)$ and $\left(ii\right)$, we get
$a^2=5,b^2=4$
$\therefore$ Equation of hyperbola is $\frac{x^2}{5}-\frac{y^2}{4}=1$
$\Rightarrow 4x^2-5y^2=20$