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4 votes
A parabola can be drawn given a focus of

(
2
,
8
)
(2,8) and a directrix of
y
=
10
y=10. Write the equation of the parabola in any form.

User Sbhklr
by
8.1k points

1 Answer

5 votes

Answer:


y=-(1)/(4)(x-2)^(2)+9

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=-(1)/(4)(x-2)^(2)\\y=-(1)/(4)(x-2)^(2)+9

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

User Kindohm
by
8.8k points

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