Question

What is the equation of a parabola satisfying the given information? Focus​ (8​, 2) directrix x= -9

Answer 1

`color(maroon)( x = y^2 / 34 - (2/17)y - 13/34`

Explanation:

Let (x0,y0) be any point on the parabola. Find the distance between (x0,y0) and the focus.

Then find the distance between (x0,y0) and directrix.

Equate these two distance equations and the simplified equation in x0 and y0 is equation of the parabola.

`"The distance between "(x_0,y_0)" and (8,2) is "sqrt((x_0−8)^2+(y_0−2)^2)`

`"The dist. bet. (x0,y0) and the directrix, y=3 is " | y_0- (-9) |`.

Equate the two distance expressions and square on both sides.

`(x_0 - 8)^2 + (y_0 - 2)^2 = (x_0 + 9)^2`

`cancel(x_0^2) - 16x_0 + 64 + y_0^2 - 4y_0 + 4 = cancel(x_0^2) + 18x_0 + 81`

`-16x_0 - 18x_0 = -y_0^2 + 4y_0 - 64 - 4 + 81`

`34y_0 = y_0^2 - 4y_0 - 13`

`x_0 = y_0^2/34 - (2/17)y_0 - 13/34`

`"This equation in " (x_0,y_0) " is true for all other values on the parabola and hence we can rewrite with " (x,y)`

`:. color(maroon)( x = y^2 / 34 - (2/17)y - 13/34`

Answer 2

`x = 1/34y^2-2/17y-13/34`

Explanation:

The distance from the focus, `(8,2)` to any point, `(x,y)` on the parabola is:

`d = sqrt((x-8)^2+(y-2)^2)" [1]"`

The distance from the directrix, `x = -9` to any point, `(x,y)` on the parabola is:

`d = x- -9`

Minus a minus is a plus:

`d = x+9" [2]"`

Because a parabola is defined as the locus of points equidistant to its focus and its directrix, we set the right side of equation [1] equal to the right side of equation [2]:

`sqrt((x-8)^2+(y-2)^2) = x+9`

Square both sides of the equation:

`(x-8)^2+(y-2)^2 = (x+9)^2`

Expand the squares:

`x^2-16x+64+y^2-4y+4=x^2+18x+81`

Combine like terms:

`-34x +y^2-4y-13=0`

Write in the standard form:

`x = 1/34y^2-2/17y-13/34`

This is a parabola that opens to the right:

graph{x = 1/34y^2-2/17y-13/34 [-19.23, 84.77, -21.25, 30.8]}