analytoc geometry problem! help

**kratos023** · January 18th 2013, 05:29 AM

Find the equation of the ellipse which crosses the point (-3,1) , and the multiply of the two distances between the point and the two focuses (foci) of the ellipse equals 6

**MarkFL** · January 18th 2013, 10:26 AM

For simplicity, let's assume the major axis lies along the x-axis, where the foci are at (-c,0) and (c,0), and $0<c$ .

If the product of the distances from the point (-3,1) to the foci is 6, then we may state:

$\sqrt{(-3-(-c))^2+(1-0)^2}\sqrt{(-3-c)^2+(1-0)^2}=6$

$\sqrt{(3-c)^2+1}\sqrt{(3+c)^2+1}=6$

Squaring gives:

$((3-c)^2+1)((3+c)^2+1)=36$

Now solve this for $c^2$ .

Next, the equation of the ellipse may be written:

$\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$

Since the ellipse passes through (-3,1), we may write:

$\frac{(-3)^2}{a^2}+\frac{1^2}{a^2-c^2}=1$ where $c^2<a^2$ .

Using the value you found for $c^2$ , you may now solve for $a^2$ to find the equation of the ellipse.

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