Question

Foci at (-1,2) and (3,2); vertex at (-7,2)Determine the standard equation of the ellipse

Answer 1

The standard equation of the ellipse is as follows:

`((x-h)^2)/(a)+((y-k)^2)/(b)=1`

where (h,k) is the coordinates of the center.

Since the y-coordinates of the foci are the same, the formula for the foci is as follows:

`(h\pm c,k)`

Thus, to obtain the value of c, equate the x-coordinates to h+c and h-c and then solve for c.

`\begin{gathered} h+c=3 \\ h-c=-1 \end{gathered}`

Add the two equations and then solve for h.

`\begin{gathered} 2h=2 \\ h=1 \end{gathered}`

Since h=1, substitute the value of h into h+c and solve for c.

`\begin{gathered} h+c=3 \\ 1+c=3 \\ c=2 \end{gathered}`

Since the vertex is (-7,2), the value of k is 2. Equate the -7 to h-a and then solve for a.

`\begin{gathered} h-a=-7 \\ 1-a=-7 \\ a=8 \end{gathered}`

Substitute the value of a and c into the following equation and then solve for b².

`\begin{gathered} a^2=b^2+c^2 \\ 8^2=b^2+2^2 \\ 64=b^2+4 \\ 60=b^2 \end{gathered}`

Obtain the value of a².

`a^2=8^2=64`

Identify the coordinates of the center.