A parabola can be drawn given a focus of (8, -1)(8,−1) and a directrix of y=-11y=−11. Write the equation of the parabola in any form.

The equation of parabola of given properties is:

Given data about parabola:

Coordinates of point of focus: (8, -1)

Equation of directrix: y = -11

The key fact in solving such problem is that each point of parabola is equally distant from the point of focus and perpendicular line on the directrix.

Let a point of parabola be

Then,  we have:

Since this is true for all (x,y), thus we get equation of parabola as:

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(x - 8)² = 20(y + 6)

Step-by-step explanation:

The focus of the parabola is (8,-1) and the equation of the directrix is y = - 11.

So, the parabola opens up and the axis of symmetry is x = 8 line.

Now, the axis of symmetry and the directrix meet at the point (8,-11).

So, the vertex of the parabola will be the midpoint of (8,-1) and (8,-11) i.e. (8,-6).

Now, half of the distance between the focus and the directrix is = a = units.

So, the equation of the parabola in vertex form will be

(x - 8)² = 4 × 5 × (y + 6)

⇒ (x - 8)² = 20(y + 6) (Answer)

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