Question

Find an equation for the parabola with focus at (-5,-4) and vertex at (-5,-3)

Answer 1

Answer:

$(x+5)^{2}=-4(y+3)$

Step-by-step explanation:

Given:

Focus point = (-5, -4)

Vertex point = (-5, -3)

We need to find the equation for the parabola.

Solution:

Since the x-coordinates of the vertex and focus are the same,

so this is a regular vertical parabola, where the x part is squared. Since the vertex is above the focus, this is a right-side down parabola and p is negative.

The vertex of this parabola is at (h, k) and the focus is at (h, k + p). So, directrix is y = k - p.

Substitute y = -4 and k = -3.

$-4 = -3+p$

$p=-4+3$

$p=-1$

So the standard form of the parabola is written as.

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

Substitute vertex (h, k) = (-5, -3) and p = -1 in the above standard form of the parabola.

So the standard form of the parabola is written as.

$(x-(-5))^{2}=4(-1)(y-(-3))$

$(x+5)^{2}=-4(y+3)$

Therefore, equation for the parabola with focus at (-5,-4) and vertex at (-5,-3)

$(x+5)^{2}=-4(y+3)$