Question

The focus of a parabola is (4,7) and the directrix is y=8.

Answer 1

We know that the x-ordinate of the vertex is x = 4, because the x-ordinate of the vertex will not change.

Now, the definition of a parabola in locus terms is that for any fixed point that lies on the parabolic shape, it is equidistant from the directrix to the fixed point and the focus to the fixed point.

Thus, we can deduce that:
PS = PD, where P(x, y) is a moving point that lies on the parabola.

Now, we know that the latus rectum will be -4a, because the directrix is higher than the y-ordinate of the focus. Thus, we can say that the vertex will be the midpoint of the two points y = 7 and y = 8, because that is the only place where the definition of a parabola works.

Hence, we can say that the vertex lies at (4, 7.5).

Thus, our general form becomes:
(x - 4)² = 4a(y - 7.5)

To find a, we can say: 7 = 7.5 + a and 8 = 7.5 - a
a = -0.5, because a is the factor in which we move to find the directrix and focus.

Thus, our equation becomes: (x - 4)² = -2(y - 7.5)