Question

Write the equation of the direction of the parabola $x^2-4x-8y+12=0$.

Answer 1

The given system of equation is $x^2-4x-8y+12=0$.

$\Rightarrow x^2-4x=8y-12$

$\Rightarrow x^2-2\times x\times 2+4=8y-12+4$

$\Rightarrow (x-2{)}^2=8y-8$

$\Rightarrow (x-2{)}^2=8(y-1)...(i)$

Shifting the origin to the point (2,1) without rotating the axes and denoting the new coordinates w.r.t. these axes by X and Y.

x=X+2,y=Y+1 ...(ii)

Using these relations,equation (i) reduces to $X^2=8Y$ ...(iii)

This is of the form $X^2=4aY,$ on comparing,we get 4a=8

$\Rightarrow a=2$

$\therefore$ equation of the directrix of the parabola w.r.t. new axes is Y=-2

$\therefore y=-2+1$ [Using equation (ii)]

$\Rightarrow y=-1$

$\Rightarrow y+1=0$

$\Rightarrow$ equation of the directrix of the parabola w.r.t. old axes is y+1=0.