Question

Which of the following equations represents an ellipse having a major axis oflength 18 and foci located at (4,7) and (4,11)?Options are pictured

Answer 1

If the x coordinate of the foci of an ellipse is the same on both foci, we call this an ellipse with a major axis parallel to the y-axis. The equation of these ellipses is:

$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$

Where:

The length of the major axis is 2a

The coordinates of the foci are: (h, k ± c), where c² = a² - b²

In this case, we can see that the x-coordinate on both foci is the same: 4. Thus, this is an ellipse with its major axis parallel to the y-axis, with a major axis length of 18, and the coordinates of the foci are (4, 7) and (4, 11)

From this, we can find a:

$\begin{gathered} 18=2a \\ . \\ a=\frac{18}{2}=9 \end{gathered}$

We can also see that h = 4. We need to find k and b.

We know:

$\begin{cases}k+c={11} \\ k-c={7}\end{cases}$

We can add these equations:

$\begin{gathered} (k+c)+(k-c)=11+7 \\ . \\ \end{gathered}$

And solve for k:

$\begin{gathered} 2k=18 \\ . \\ k=\frac{18}{2}=9 \end{gathered}$

Now that we know k = 9, we can find c:

$\begin{gathered} 9+c=11 \\ . \\ c=11-9=2 \end{gathered}$

And finally, use the formula to find b:

$\begin{gathered} 2^2=9^2-b^2 \\ . \\ b^2=81-4 \\ . \\ b=\sqrt{77} \end{gathered}$

We have all the needed values:

$\begin{gathered} h=4 \\ k=9 \\ a=9 \\ b=\sqrt{77} \end{gathered}$

We can write:

$\frac{(x-4)^2}{77}+\frac{(y-9)^2}{81}=1$

The correct answer is option A.