Question

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

Answer choices:
a) f(x) = 1/32 (x+7)^2-3
b) f(x) = -1/32 (x+7)^2-3
c) f(x) = -1/32 (x-7)^2 -3
d) f(x) = 1/32 (x-7)^2 -3

Answer 1

Please, if (x,y) is a point of your parabola, what is the distance between (x,y) and the focus (-7,5)?

Answer 2

(-7, -6 ) ?

Answer 3

oops wait i mean ( -7, -3 )?

Answer 4

you ahve to apply this formula, distance d is:
\[d=\sqrt{(x _{2}-x _{1})^{2}+(y _{2}-y _{1})^{2}}\]
where (x_1,y_1) is the initial point and (x_2,y_2) is the focus

Answer 5

namely, (x_1,y_1)=(x,y) and (x_2,y_2)=(-7,5).
Please substitute them into the formula for d

Answer 6

for example, we have:
\[x _{2}-x _{1}=x-(-7)=x+7\]
and:
\[y _{2}-y _{1}=y-5\]
so, please substitute them

Answer 7

ok so (x+7)^2 = x^2 + 14x + 49 and (y-5)^2 = y^2 -10y +25

Answer 8

ok!, so:
\[d=\sqrt{x ^{2}+49+14x+y ^{2}+15-10y}\]

Answer 9

is the answer x^2 + y^2 + 14x - 10y _ 64?

Answer 10

Now, what is the distance between the point (x,y) and the directrix? please note this drawing: