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 $\displaystyle (-1, 3)$. This means the equation will be of the form:
$\displaystyle \frac{(x+1)^2}{a^2}+\frac{(y-3)^2}{b^2}=1$
The axis is along the line $\displaystyle x = -1$, so the major axis is vertical, having foci where $\displaystyle y = 3 \pm be$, $\displaystyle b$ being the length of the semi-major axis and $\displaystyle e$ the eccentricity. This gives:
$\displaystyle be = 4$
The length of the semi-minor axis is the value of $\displaystyle a$. So $\displaystyle a = 8$.

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