Question

The parametric equations of a parabola are x = t² + 1, y = 2t + 1. The cartesian equation of its directrix is
(a) x = 0
(b) x + 1 = 0
(c) y = 0
(d) none of these

Answer 1

(a) x = 0

Given:
x = t² + 1 (1)
y = 2t + 1 (2)

From (1) and (2):
$x={\left(\frac{y-1}{2}\right)}^2+1$

On simplifying:
${\left(y-1\right)}^2=4\left(x-1\right)$

Let $Y=y-1\mathrm{and}X=x-1$

∴ $Y^2=4X$

Comparing it with y² = 4ax:
a = 1

Therefore, the equation of the directrix is X = −a , i.e. $x-1=-1\Rightarrow x=0$.