Question

What is the equation of a parabola with (4, 6) as its focus and y=2 as its directrix?

a) (x−4)² =8(y−6)
b) (x−4)² =4(y−2)
c) (x−4)² =2(y−6)
d) (x−4)² =6(y−2)

Answer 1

The equation of the parabola with focus (4, 6) and directrix y=2 is (x−4)² = 8(y−6), which corresponds to option a).

Step-by-step explanation:

The equation of a parabola can be found using the focus and directrix. The standard form of the equation of a vertically oriented parabola with focus (h, k) and directrix y = d is:

(x-h)^2 = 4p(y-k)

Where p is the distance between the focus and directrix. Here, the focus is (4, 6) and the directrix is y = 2, so 'h' is 4, 'k' is 6, and 'p' is the distance from the focus to the directrix, which is 6 - 2 = 4.

So we substitute 'h', 'k', and 4p into the formula:

(x-4)^2 = 4*4(y-6)

(x-4)^2 = 16(y-6)

To match the options given in the question, we can simplify by dividing both sides by 2:

(x-4)^2 = 8(y-6)

Therefore, the correct equation of the parabola is option a) (x−4)² = 8(y−6).