Question

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

Answer 1

lots of ways to do this question



you can use the distance formula. P is a point (x, y) on the parabola

and then use the fact the the distance from the focus to P is the same as the distance from P to a point on the directrix D

so

\[\sqrt{x + 7)^2 + (y - 5)^2} = \sqrt{(x - x)^2 + (y +11)^2}\]

the alternative is to find the focal distance = a

this the the distance from the vertex to the focus..

the distance from the focus to the directrix is 2a

so 2a = 16

then the vertex is a units below the focus on the same line of symmetry x = -7

so the vertex is (-7, 5 - a)

then substitute the values into the standard form

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

where (h, k) is the focus and a is the focal length.

hope it helps