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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

Respuesta :

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 [tex](x-h)^2=4p(y-k)[/tex] then the focus of parabola is defined by [tex]f(h,k+p)[/tex] and directrix is defined by [tex]y=k-p[/tex].

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

[tex]k+p=5[/tex]    ...(1)

The given directrix is [tex]y=-11[/tex],

[tex]k-p=-11[/tex]   ....(2)

Add equation (1) and (2),

[tex]2k=-6\\k=-3[/tex]

Put [tex]k=-3[/tex] in equation (1), we get [tex]p=8[/tex].

Substitute these values in equation [tex](x-h)^2=4p(y-k)[/tex].

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

Therefore, the correct option is option (1).

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