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Paul [167]
3 years ago
10

A parabola can be drawn given a focus of

Mathematics
1 answer:
Volgvan3 years ago
4 0

Answer:

y=-\frac{1}{4}(x-2)^{2}+9

Step-by-step explanation:

Any point on a given parabola is equidistant from focus and directrix.

Given:

Focus of the parabola is at (2,8).

Directrix of the parabola is y=10.

Let (x,y) be any point on the parabola. Then, from the definition of a parabola,

Distance of (x,y) from focus = Distance of (x,y) from directrix.

Therefore,

\sqrt{(x-2)^{2}+(y-8)^{2}}=|y-10|

Squaring both sides, we get

(x-2)^{2}+(y-8)^{2}=(y-10)^{2}\\(x-2)^{2}=(y-10)^{2}-(y-8)^{2}\\(x-2)^{2}=(y-10+y-8)(y-10-(y-8))...............[\because a^{2}-b^{2}=(a+b)(a-b)]\\(x-2)^{2}=(2y-18)(y-10-y+8)\\(x-2)^{2}=2(y-9)(-2)\\(x-2)^{2}=-4(y-9)\\y-9=-\frac{1}{4}(x-2)^{2}\\y=-\frac{1}{4}(x-2)^{2}+9

Hence, the equation of the parabola is y=-\frac{1}{4}(x-2)^{2}+9.

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