Question

Write the coordinates of the vertex of the parabola whose focus is at (−2, 1) and directrix is the line x + y − 3 = 0.

Answer 1

Given:
The focus S is at (−2, 1) and the directrix is the line x + y − 3 = 0.

The slope of the line perpendicular to x + y − 3 = 0 is 1.

The axis of the parabola is perpendicular to the directrix and passes through the focus.

∴ Equation of the axis of the parabola = $y-1=1\left(x+2\right)$ (1)

Intersection point of the directrix and axis is the intersection point of (1) and x + y − 3 = 0.

Let the intersection point be K.

Therefore, the coordinates of K are (0, 3).

Let (h, k) be the coordinates of the vertex, which is the mid-point of the line segment joining K and the focus.

$$\begin{gathered} \therefore h=\frac{0-2}{2},k=\frac{3+1}{2} \\ h=-1,k=2 \end{gathered}$$

Hence, the coordinates of the vertex are (−1, 2).