Question

How do you find the foci of an ellipse not centered at the origin $?$

Answer 1

Step 1: Writing the equation of an ellipse:

Let $(h,k)$ be the centre of an ellipse. $h$ and $k$ define the position of the ellipse, where $“h”$ units to the left or right and $“k”$ units up or down ellipse move on the axes.

The standard equation of the ellipse at centre $(h,k)$ will be

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

Step 2: Finding foci of an ellipse:

Now, to find the foci, use the formula ${\mathit{c}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}{\mathit{a}}^{\mathbf{2}}\mathbf{}\mathbf{–}\mathbf{}{\mathit{b}}^{\mathbf{2}}$

The foci of the ellipse depend on the axis of the ellipse.

For horizontal, the equation is:

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

Foci for horizontal $=(h\pm c,k)$

For vertical, the equation is:

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

Foci for vertical $=(h,k\pm c)$

Hence, in this way, we can find the foci of an ellipse not centred at the origin.