Question

Equation of one of the tangents passing through $(2,8)$ to the hyperbola $5x^2-y^2=5$ is

• A. $3x-y-2=0$
• B. $23x-3y-22=0$
• C. $x+y+3=0$
• D. $10x-8y-5=0$

Answer 1

The correct option is B $23x-3y-22=0$

Equation of the hyperbola is $5x^2-y^2=5\Rightarrow \frac{x^2}{1}-\frac{y^2}{5}=1$
Here, $a^2=1$ and $b^2=5$
Equation of the tangent is $y=mx\pm \sqrt{a^2{m}^2-b^2}$ and it passes through $(2,8)$
$\therefore 8=2m\pm \sqrt{m^2-5}$
$\Rightarrow (8-2m{)}^2=m^2-5$
$\Rightarrow 64+4m^2-32m=m^2-5$
$\Rightarrow 3m^2-32m+69=0$
$\Rightarrow 3m^2-9m-23m+69=0$
$\Rightarrow (3m-23)(m-3)=0$
$\therefore m=\frac{23}{3},3$
Therefore, equation of required tangent's are
$y=3x+2$ and $23x-3y-22=0$