Question

If the focus of a parabola is (−2, 1) and the directrix has the equation x + y = 3, then its vertex is
(a) (0, 3)
(b) (−1, 1/2)
(c) (−1, 2)
(d) (2, −1)

Answer 1

(c) (−1, 2)
Given:
The focus S is at (−2, 1) and the directrix is the line x + y − 3 = 0......(1)
The slope of the line perpendicular to $y=-x-3$ is $-1$.
The axis of the parabola is perpendicular to the directrix and passes through the focus.
∴ Equation of the axis of the parabola = $y-1=1(x+2)$
$\Rightarrow y-1=x+2$
$\Rightarrow x-y+3=0$............(2)
Intersection point of the directrix and the axis is the intersection point of equation (2) and equation (1).
Adding equation (1) and (2), we get
$\Rightarrow 2x=0$
$\therefore x=0$
Substitute x=0 in equation(1), we get
$\Rightarrow y=3$
Therefore, the coordinates will be K(0, 3).
Let (h, k) be the coordinates of the vertex, which is the mid-point of the segment joining K and the focus.
Thus, $h=\frac{0-2}{2},k=\frac{3+1}{2}$
$\Rightarrow h=\frac{-2}{2},k=\frac{4}{2}$
$h=-1,k=2$
Hence, the coordinates of the vertex are (−1, 2).