Chapter 10.3, Problem 34E

To determine

To write: The equation of the ellipse whose semi-major axis has length $2\sqrt{13}$ units and the foci are at $\left(-1,1\right)\text{ and}\left(-1,-5\right)$ .

Answer to Problem 34E

  $\frac{{\left(x+1\right)}^2}{43}+\frac{{\left(y+2\right)}^2}{52}=1$

Explanation of Solution

Given information: Semi-major axis has length $2\sqrt{13}$ units and the foci are at $\left(-1,1\right)\text{ and}\left(-1,-5\right)$

Calculation:

The foci are vertical, so then is the major axis, giving us:

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

The center is between the foci $\left(-1,1\right),\left(-1,-5\right)$ . This puts the center at $\left(-1,-2\right),$ giving us:

  $\frac{{\left(x+1\right)}^2}{b^2}+\frac{{\left(y+2\right)}^2}{a^2}=1$

The length of the semi-major axis,

  $a=2\sqrt{13}\to a^2=52$

And we know from the foci points, that c = 3.

  $52-b^2=9\to b^2=43$

Giving final equation:

  $\frac{{\left(x+1\right)}^2}{43}+\frac{{\left(y+2\right)}^2}{52}=1$