# A radio telescope has a parabolic dish with diameter of 100 metres. The collected radio signals are reflected to one collection point, called the "focal" point, being the focus of the parabola, If the focal length is 45 metres, find the depth of the dish, rounded to one decimal place.

For simplicity, I'll center the curve for the arch on the y-axis, so the vertex will be at(h, k) = (0, 25). Since the width is thirty, then the x-intercepts must be at x = –15 and x = +15. Obviously, this is a regular (vertical) but upside-down parabola, so the x part is squared and I'll have a negative leading coefficient.

Working backwards from the x-intercepts, the equation has to be of the form y =
a(x – 15)(x + 15). Plugging in the known vertex value, I get:

Then a = –1/9. With a being the leading coefficient from the regular quadratic equation y = ax2+ bx + c, I also know that the value of 1/a is the same as the value of 4p, so 1/(–1/9) = –9 = 4p, and thus p = –9/4.

The focus is 9/4 units below the vertex; the directrix is the horizontal line 9/4 units above the vertex:

​25 = a(0 – 15)(0 + 15) = –225a
4p(y – h) = (x – k)2

4(–9/4)(y – 25) = (x – 0)2
–9(y – 25) = x2
focus: (0, 91/4), directrix: y = 109/4
You could also work directly from the conics form of the parabola equation, plugging in the vertex and an x-intercept, to find the value of p:

4p(y – 25) = (x – 0)2
4p(0 – 25) = (15 – 0)2
4p(–25) = 225
4p = –225/25 = –9
p=-9/4

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