Question

Regan is trying to find the equation of a quadratic that has a focus of (-2, 5) and a directrix of y = 13. Describe to Regan your preferred method for deriving the equation. Make sure you use Regan's situation as a model to help her understand.

Answer 1

Quadratics are parabolas. One of the definitions of parabola is that it is a curve where the distance from a point on the curve to the focus is the same as the distance from the point to the directrix.

So let the point be (x,y)
Find the distance between (x,y) and the focus (-2,5) using the formula for the distance between two points.

Find the distance between (x,y) and the line y = 13. It will be just |y - 13|

Equate the two distances and simpligy to get the equation of the parabola.

Answer 2

What is the x and y values?

Answer 3

It is a generic point on the parabola. No specific point. When you find the distances and equate them you will get the equation of the parabola in the form: y = ax^2 + bx + c

Answer 4

so how do i do the first step?

Answer 5

All points on the parabola are equidistant from the focus and the directrix
distance from focus d^2 = distance from directrix d1^2
(x + 2)^2 + (y - 5)^2 = (13 - y)^2
Expand and simplify.

Answer 6

thank you

Answer 7

you are welcome.

Answer 8

@ranga im having trouble simplifying it. what is the final answer?

Answer 9

x2+4x+4+y2-10y+25=169-26y+y2. @ranga this is what i have

Answer 10

Simplifying I am getting:

y = -1/16x^2 - 1/4x + 35/4

Answer 11

When plotted t will be an inverted parabola (an upside down U) because the leading coefficient is negative (-1/16)

Answer 12

how do you simply what i got into what you got?

Answer 13

simplify

Answer 14

x^2 + 4x + 4 + y^2 - 10y + 25 = 169 - 26y + y^2
subtract y^2 from both sides
x^2 + 4x + 4 - 10y + 25 = 169 - 26y
subtract 169 from both sides
x^2 + 4x - 10y + 4 + 25 - 169 = - 26y
x^2 + 4x - 10y - 140 = - 26y
Add 10y to both sides
x^2 + 4x - 140 = -16y
divide both sides by -16
-1/16x^2 - 4/16x + 140/16 = y
y - -1/16x^2 - 1/4x + 35/4

Answer 15

thanks

Answer 16

yw.