Question

The directrix of the parabola x² − 4x − 8y + 12 = 0 is

Options

• y = 0 

• x = 1 

•  y = − 1 

• x = − 1 

Answer 1

 y = −1 

Given:
x² − 4x − 8y + 12 = 0  

\[\Rightarrow \left( x - 2 \right)^2 - 4 - 8y + 12 = 0\]
\[ \Rightarrow \left( x - 2 \right)^2 = 8y - 8\]
\[ \Rightarrow \left( x - 2 \right)^2 = 8\left( y - 1 \right)\] 

Putting X = x − 2, Y = y − 1: 

\[X^2 = 8Y\] Comparing with \[X^2 = 4aY\] a = 2

Equation of the directrix: \[Y = - a\] 

⇒ \[Y = - 2\] 

\[\Rightarrow y - 1 = - 2\]
\[ \Rightarrow y = - 2 + 1\]
\[ \Rightarrow y = - 1\]