Question

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

Answer 1

what is half way between \((-7,5)\) and \(y=-11\)? that is the vertex and we need that first

Answer 2

should be clear that it is the point \((-7,-3)\) since \(\frac{5-11}{2}=-3\)

Answer 3

then use
\[(x-h)^2=4p(y-k)\] with
\[h=-7,k=-3\] and \(p\) is the distance between the vertex and the focus

Answer 4

also hope it is clear that that distance is 8
so plug in the numbers and you are done
we can check it if you like

Answer 5

yw
here is the check
http://www.wolframalpha.com/input/?i=parabola+%28x%2B7%29^2%3D32%28y%2B3%29

Answer 6

For any given focus \((x_1, y_1\)) and directrix \((x_2, y_2)\), such points can be inserted in to the formula:

\((x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2\)

And simplified to produce the equation of the parabola.

In this case you are given the focus\((-7, 5)\) and directrix \((x, -11)\). Inserting those points into the formula, you get:
\((x - (-7))^2 + (y - 5)^2 = (x - x)^2 + (y - (-11))^2\)

Which simplifies to
\((x + 7)^2 + (y - 5)^2 = (y + 11)^2\)

And expands to

\(x^2 + 14x + 49 + y^2 - 10y + 25 = y^2 + 21y + 121\)

Notice that \(y^2\) cancels on both sides which reduces the equation to
\(x^2 + 14x + 49 -10y + 25 = 21y + 121\)
\(x^2 + 14x + 49 + 25 - 121 = 21y + 10y\)
\((x + 7)^2 - 96 = 31y\)

Divide both sides by 31 to isolate \(y\)

Answer 7

I made a mistake.

Answer 8

You expand to get
\(x^2 + 14x + 49 + y^2 - 10y + 25 = y^2 + 22y + 121\)

\(y^2\) cancels to get:
\(x^2 + 14x + 49 -10y + 25 = 22y + 121\)
Which simplifes to
\((x + 7)^2 - 96 = 32y\)

Divide both sides by 32 to get
\(\dfrac{(x + 7)^2}{32} - 3 = y\)

Answer 9

wow !!

Answer 10

It's easier than it looks @satellite73

Answer 11

I'm not as fast as you with the Latex

Answer 12

where in the world does this come from \[(x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2\]?

Answer 13

looks like the distance formula ?

Answer 14

Yes

Answer 15

i see
but you don't need a distance formula when the two points are on the same vertical or horizontal axis
i.e. i don't need the distance formula to find the distance between \((-7,5)\) and \((-7,-11)\) it is just 16 by your eyeballs

Answer 16

not that it is wrong, it is just a lot of algebra for not much payoff imho

Answer 17

I can do it much quicker by hand without the latex. It's not that much work. I was just showing the detailed steps.

Answer 18

It's not that difficult to expand binomial squares.

Answer 19

You prefer the graphical approach. I prefer the algebraic approach

Answer 20

i agree, but i can't do it in my head
i can find the distance between any two points on the same vertical or horizontal in my head though

Answer 21

Most students prefer the algebraic approach though and avoid graphical methods at all costs.