Equation for ellipses not centred at Origin

The equation of an ellipse that is offset from (0,0) is very similar to the standard form, except that the centre coordinates (p,q) are subtracted from the variables x and y respectively. That is...

\frac{\left ( x-p \right )^2}{a^2}-\frac{\left ( y-q \right )^2}{b^2}=1

Here, a and b still are the lengths of the semi-major and semi-minor axes respectively.

For an ellipse with a given semi-major and semi-minor axis, the shape and every other feature remains the same as one located at (0,0), except all features have been offset by the centre point coordinates (p, q). That is...

The centre is located at: (p,q) rather than (0,0)
The foci are located at: (±ae+p , q) rather than (±ae , 0)
The horizontal vertices are located at: (±a+p , q) rather than (±a , 0)
The vertical vertices are located at: (±p, b+q) rather than (0 , ±b)
The directrices have the equations: x = ±(a/e)+p rather than x = ±a/e

In this video, we also sketch the ellipse formed by the equation:

9x^2 + 36x + 16y^2 + 96y = -36

Thanks for watching. Please give me a "thumbs up" if you have found this video helpful.

Please ask me a mathematics question by commenting below and I will try to help you in future videos.