Question

The directrix of the parabola x² − 4x − 8y + 12 = 0 is
(a) y = 0
(b) x = 1
(c) y = − 1
(d) x = − 1

Answer 1

(c) y = −1

Given:
x² − 4x − 8y + 12 = 0
$$\begin{gathered} \Rightarrow {\left(x-2\right)}^2-4-8y+12=0 \\ \Rightarrow {\left(x-2\right)}^2=8y-8 \\ \Rightarrow {\left(x-2\right)}^2=8\left(y-1\right) \end{gathered}$$

Putting X = x − 2, Y = y − 1:

$X^2=8Y$

Comparing with $X^2=4aY$:
a = 2

Equation of the directrix:
$Y=-a$
⇒ $Y=-2$
$$\begin{gathered} \Rightarrow y-1=-2 \\ \Rightarrow y=-2+1 \\ \Rightarrow y=-1 \end{gathered}$$