Write an equation of the ellipse with center at the origin, a vertex at (5,0) and a focus at (-4,0)

Step-by-step explanation:

For this case we have the following info:

Vertex =(5,0) = (h,k) let be the coordinates

Focus=(-4,0)=(h+p ,k) let be the coordinates

On this case since the y coordinates is always 0 we know that the parabola is on the x axis and k=0. Since the coordinate for th focus is less than the coordinate for the vertex we know that the parabola open to the left of th x axis.

And the general equation for a parabola with opens on the x axis is given by this equation:

Since k =0 then the value for h is given by the vertex on this case h=5, and we have this:

We can use the point of the focus in order to find the value of p who represent the value in order to find the directirx line for the parabola. If we replace (-4,0) into the equation we got:

h+p = -4 , 5+p =-4 so then p -4-5 = -9, and makes sense since the parabola opens to the left.

So then our equation would be given by:

2nd one in the first row is correct