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
For simplicity, let's assume the major axis lies along the x-axis, where the foci are at (-c,0) and (c,0), and $\displaystyle 0<c$.
If the product of the distances from the point (-3,1) to the foci is 6, then we may state:
$\displaystyle \sqrt{(-3-(-c))^2+(1-0)^2}\sqrt{(-3-c)^2+(1-0)^2}=6$
$\displaystyle \sqrt{(3-c)^2+1}\sqrt{(3+c)^2+1}=6$
Squaring gives:
$\displaystyle ((3-c)^2+1)((3+c)^2+1)=36$
Now solve this for $\displaystyle c^2$.
Next, the equation of the ellipse may be written:
$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$
Since the ellipse passes through (-3,1), we may write:
$\displaystyle \frac{(-3)^2}{a^2}+\frac{1^2}{a^2-c^2}=1$ where $\displaystyle c^2<a^2$.
Using the value you found for $\displaystyle c^2$, you may now solve for $\displaystyle a^2$ to find the equation of the ellipse.
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