Question

Find the equation of the tangent to the hyperbola $x^2-4y^2=36$ which is perpendicular to the line $x-y+4=0$

Answer 1

Let m be the slope of the tangent. Since the tangent is perpendicular to the line $x-y+4=0$

$\therefore m\times \left(1\right)=-1$ $\Rightarrow m=-1$

Since, $x^2-4y^2=36$ or $\frac{x^2}{36}-\frac{y^2}{9}=1$

Comparing this with

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\therefore$ $a^2=36$ and $b^2=9$

So the equations of tangents are,

$y=(-1)x\pm \sqrt{36\times (-1{)}^2-9}$

$y=-x\pm \sqrt{27}$

$x+y\pm 3\sqrt{3}=0$