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.

\[\therefore h = \frac{0 - 2}{2}, k = \frac{3 + 1}{2}\]
\[h = - 1, k = 2\]  

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