Derive the Equation of the Parabola with a Focus at (2, 4) and a Directrix of y = 8.
We will be solving this by using the focus point and the directrix.
Answer: The Equation of the Parabola with a Focus at (2, 4) and a Directrix of y = 8 is x2 - 4x + 8y - 44 = 0.
Let us solve this step by step.
Explanation:
Given that, Focus at (2, 4) and a directrix of y = 8
Let us suppose that there is a point (x, y) on the parabola.
Its distance from the focus point (2, 4) is √(x - 2)2 + (y - 4)2
Its distance from directrix y = 8 is |y - 8|
Therefore, the equation will be:
√(x - 2)2 + (y - 4)2 = |y - 8|
Apply square on both sides.
(x - 2)2 + (y - 4)2 = (y - 8)2
x2 - 4x + 4 + y2 - 8y + 16 = y2 - 16y + 64
x2 - 4x + 8y - 44 = 0