Question

Find the equations of the tangents to the hyperbola $3x^2-4y^2=12$, which are:
(i) Parallel and (ii) Perpendicular to the line
$y=x-7$

Answer 1

An equation of a tangent to a hyperbola is given by $y=mx\pm \sqrt{a^2{m}^2-b^2}$

Here, $a^2=4$ and $b^2=3$

The equation thus becomes $y=mx\pm \sqrt{4m^2-3}$

(i) Since we need a tangent parallel to the equation $y=x-7$, the value of m would be 1.

$\therefore$ the equation becomes $y=x\pm \sqrt{4-3}$ i.e. $y=x\pm 1$

(ii) A tangent perpendicular to the line $y=x-7$ implies the value of m would be $-1$

The equation thus becomes $y=-x\pm 1$