Question

The directrix of the parabola ${x^2} - 4x - 8y + 12 = 0$ is:
$
  A) y = 0 \\
  B) x = 1 \\
  C) y = - 1 \\
  D) x = - 1 \\
 $

Answer 1

Hint: A parabola is a set of all points in a plane which are equal distance away from a given point and line. The point is called the focus of the parabola, and the line is called the directrix. The directrix is perpendicular to the axes of symmetry of a parabola and does not touch the parabola.
The standard equation of the parabola is${\left( {x - b} \right)^2} = 4p\left( {y - k} \right)$, where ($h,k + p$) is the focus and $y = k - p$ is the directrix. Here, in the question the question of a parabola is given and we need to find its directrix. However the equation is not in its standard form so first we convert it into standard form.
In this question, we need to determine the directrix of the parabola which can be done by comparing the given equation with the standard equation of the parabola.

Complete step-by-step solution:
The equation of the parabola is ${x^2} - 4x - 8y + 12 = 0$
First, we convert this equation in standard form as: ${x^2} - 4x = 8y - 12$
Adding 4 to both sides –
$
  \Rightarrow {x^2} - 4x + 4 = 8y - 12 + 4 \\
\Rightarrow {x^2} - 2 \times 2x + {\left( 2 \right)^2} = 8y - 8 \\
\Rightarrow {\left( {x - 2} \right)^2} = 8\left( {y - 1} \right) \\
\Rightarrow {\left( {x - 2} \right)^2} = 4 \times 2\left( {y - 1} \right) - - - - (i) \\
 $
Equation (i) is in the form ${\left( {x - h} \right)^2} = 4p\left( {y - k} \right)$
where, $h = 2$,$p = 2$,$k = 1$
Hence, from the equation it is clear that it is a parabola opening upward whose vertex is at $\left( {2,1} \right)$ and focuses at $\left( {2,3} \right)$. [As vertex is$\left( {p,k} \right)$and focus is$\left( {h,k - p} \right)$],
So the equation of the directrix is given as:
$
  y = k - p \\
   = 1 - 2 \\
   = - 1 \\
 $
Hence the correct answer is option (c).
Note: If the parabola whose standard form is ${\left( {x - h} \right)^2} = 4p\left( {y - k} \right)$with focus $\left( {h,k + p} \right)$and directrix $y = k - p$is rotated, the equation of parabola becomes ${\left( {y - k} \right)^2} = 4p\left( {x - h} \right)$where focus is $\left( {h + p,k} \right)$and the directrix is$x = h - p$.