Find an equation for the parabola described by its vertex, directrix, and focus point:

Question

Find an equation for the parabola described by its vertex, directrix, and focus point: vertex (√2, -√3), directrix x=2√2, focus (0, -√3)

Answer 1

To make the problem easier to visualize plot the given information on an xy-coordinate system.  We see the vertex is in the fourth quadrant, the focus is to the left and the directrix is a vertical line at x = 2√2.  We know that the parabola cannot intersect the directrix and that the focus lies on the inside of the parabola.  The parabola is horizontal and points to the left.  The standard form for a horizontal parabola point to the left is

 

x - h = -4p(y - k)² where (h,k) is the vertex and p is the distance from the vertex to to focus

 

From the given vertex h = √2 and k = -√3

 

p = |0 - √2| = √2

 

The equation is

 

x - √2 = -4(√2)(y - (-√3))²

x - √2 = -4√2(y + √3)²