analytic geometry about ellipses

**fishlord40** · February 14th 2010, 11:33 PM

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.

**Grandad** · February 15th 2010, 12:38 AM

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.

Grandad

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