Question

Derive the equation of the parabola with a focus at (−2, 4) and a directrix of y = 6. Put the equation in standard form.

Answer 1

ok... so the parabola is concave down since the directrix is above the focus...

the distance between the focus and dirextrix is 2 units....

half this distance is the focal length.... so the focal length a = 1

the vertex is at (-2, 4 + a)

so the vertex is at (-2, 5)

so now using a standard for or the parabola for concave down

\[(x - h)^2 = -4a(y - k)\]

which has a vertex at (h, k) and a is the focal length, you can say


\[(x - (-2))^2 = 4 \times 1 \times(y - 5)\]

just simplify for the equation....

hope it helps.

Answer 2

the alternate method is to use the distance formula



pick a point on the parabola call it P(x, y)

the perpendicular from P to the directrix has coordinates (x, 6)

so then the distance from (x, 6) to P(x, y) has to equal the distance from
P(x, y) to the focus (-2, 4)

so using the distance formulas, you get

\[\sqrt{(x - x)^2 + (y - 6)^2} = \sqrt{(x - (-2))^2 + (y - 4)^2}\]

sosimplify this to find the equation...

hope it helps