Question

If the coordinates of the vertex and the focus of a parabola are (−1, 1) and (2, 3) respectively, then the equation of its directrix is
(a) 3x + 2y + 14 = 0
(b) 3x + 2y − 25 = 0
(c) 2x − 3y + 10 = 0
(d) none of these.

Answer 1

(a) 3x + 2y + 14 = 0

Given:
The vertex and the focus of a parabola are (−1, 1) and (2, 3), respectively.

∴ Slope of the axis of the parabola = $\frac{3-1}{2+1}=\frac{2}{3}$

Slope of the directrix = $\frac{-3}{2}$

Let the directrix intersect the axis at K (r, s).

∴ $$\begin{gathered} \frac{r+2}{2}=-1,\frac{s+3}{2}=1 \\ \Rightarrow r=-4,s=-1 \end{gathered}$$

Equation of the directrix:
$\left(y+1\right)=\frac{-3}{2}\left(x+4\right)$
$\Rightarrow 3x+2y+14=0$