1. ## analytic geometry about ellipses

Give the equation of the ellipse having foci at (-1, -1) and (-1, 7) and the length of the semi - minor axis is 8 units.

2. Hello fishlord40
Originally Posted by fishlord40
Give the equation of the ellipse having foci at (-1, -1) and (-1, 7) and the length of the semi - minor axis is 8 units.
The centre of the ellipse is at the mid-point of the line joining the foci. So that's at $(-1, 3)$. This means the equation will be of the form:
$\frac{(x+1)^2}{a^2}+\frac{(y-3)^2}{b^2}=1$
The axis is along the line $x = -1$, so the major axis is vertical, having foci where $y = 3 \pm be$, $b$ being the length of the semi-major axis and $e$ the eccentricity. This gives:
$be = 4$
The length of the semi-minor axis is the value of $a$. So $a = 8$.

Finally, with an ellipse with a vertical major axis, we have:
$e^2 = 1-\frac{a^2}{b^2}$
Eliminate $e$ to find $b$, and hence the equation of the ellipse.