Question

Derive the equation of the parabola with a focus at (4, −7) and a directrix of y = −15. Put the equation in standard form.

Answer choices:
a) f(x) = 1/16 x^2 -8x +11
b) f(x) = 1/16 x^2 -8x -10
c) f(x) = 1/16 x^2 -1/2x +11
d) f(x) = 1/16 x^2 -1/2x -10

Answer 1

11?

Answer 2

I mean 4,11?

Answer 3

Please, if I apply the definition of a parabola, namely "a parabola is a curve for which the distance between its point and focus is equal to the distance between that same point and the directrix", I can write:
\[(x-4)^{2}+(y+7)^{2}=(y+15)^{2}\]
Now, please continue the computation above

Answer 4

Please note that I called (x,y) a generic point which belongs to our parabola

Answer 5

lolwhat how do u solve that

Answer 6

for example:
\[(x-4)^{2}=x ^{2}+16-8x\]
please do the same with the remaining terms

Answer 7

ah ok so we're foiling them ?

Answer 8

we have to develop the squares

Answer 9

(y+7)^2= y^2 + 14y + 49 ?

Answer 10

(y+15)^2 = y^2 + 30y + 225?

Answer 11

that's right!

Answer 12

yay :)

Answer 13

what do I do from here?

Answer 14

I insert your result into my first equation, namely:
\[x ^{2}+16-8x+y ^{2}+49+14y=y ^{2}+225+30y\]
Now, please simplify

Answer 15

are we solving for x or y

Answer 16

we have to solve for y

Answer 17

does y= x^2 over 16 - x/2 -10 ?

Answer 18

perfect! Well done! :)

Answer 19

:D

Answer 20

how do I find my answer with this?

Answer 21

please note that your "y" stands for "f(x)", so?

Answer 22

yeah but there is no x^2 over 16 in the answer choices :(?

Answer 23

Please, note that your option is d), namely the fourth option

Answer 24

since 1/16 x^2 is the same as x^2 over 16

Answer 25

Oh I see now thank you very much

Answer 26

thank you!