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Derive the equation of the parabola with a focus at (−7, 5) and a directrix of y = −11.


f(x) = one thirty second (x + 7)2 − 3


f(x) = −one thirty second (x + 7)2 − 3


f(x) = −one thirty second (x − 7)2 − 3


f(x) = one thirty second (x − 7)2 − 3

Verified Answer :

meerkat18

In this problem, given the focus at (-7,5) and directrix at y = -11. then it is implied that the parabola is facing upwards. The vertex hence is at the middle of the focus and the directrix, hence at (-7, -3). The general formula of the parabola is y-k = 4a  ( x-h)^2. SUbstituting, y + 3 = 1/30 *(x+7)^2. Answer i sA.

Solution: The correct of option (1).

Explanation:

if the parabola is defined by the equation \((x-h)^2=4p(y-k)\) then the focus of parabola is defined by \(f(h,k+p)\) and directrix is defined by \(y=k-p\).

The focus is (-7,5), therefore h=-7 and

\(k+p=5\)    ...(1)

The given directrix is \(y=-11\),

\(k-p=-11\)   ....(2)

Add equation (1) and (2),

\(2k=-6\\k=-3\)

Put \(k=-3\) in equation (1), we get \(p=8\).

Substitute these values in equation \((x-h)^2=4p(y-k)\).

\((x+7)^2=4(8)(y+3)\\y=\frac{1}{32}(x+7)^2-3\)

Therefore, the correct option is option (1).

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