Derive the Equation of the Parabola with a Focus at (2, 4) and a Directrix of y = 6.
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 = 6 is x^{2} - 4x + 4y - 16 = 0.
Let us solve this step by step.
Explanation:
Given: Focus at (2, 4) and a Directrix of y = 6
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 = 6 is |y - 6|
Therefore, the equation will be:
√(x - 2)2 + (y - 4)2 = |y - 6|
Squaring on both the sides,
(x - 2)2 + (y - 4)2 = (y - 6)2
x2 - 4x + 4 + y2 - 8y + 16 = y2 - 12y + 36
x2 - 4x + 4y - 16 = 0