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

Answer:

$y = - \frac{1}{4} {(x + 2)}^{2} + 5$

Step-by-step explanation:

The vertex of this parabola is the midpoint of the focus (-2,4) and where the directrix intersects the axis of symmetry of the parabola (-2,6)

This parabola must open downwards due to the position of the directrix and has equation of the form:

${(x - h)}^{2} = - 4p(y - k)$

where (h,k) is the vertex.

This implies that:

$h = - 2$

and

$k = \frac{4 + 6}{2} = 5$

The value of p is the distance from the vertex to the focus:

$p = |6 - 5| = 1$

We substitute all the values into the formula to get:

$(x - - 2)^{2} = - 4(1){(y - 5)}$

${(x + 2)}^{2} = - 4(y - 5)$

Or

$y = - \frac{1}{4} {(x - 5)}^{2} + 5$