Finding Ellipse equation from 3 points

Question

I have 3 points

f1: (6,0)      // Foci point
f2: (-6,0)     // Foci point
p:  (8.1, 4.7) // point on the ellipse

Ellipse equation is: $$ \frac{x^{2}}{a^2}+\frac{y^{2}}{b^2}=1 $$

How do I find $ a $ and $b$?

Answer 1

Hint:

use the fact that if the foci of the ellipse are $F=(\pm c,0)$ than we have $b^2+c^2=a^2$. So you have only one free parameter in the equation that can be determined using the coordinates of the given point.

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e have $c=6$, so: $a^2=36+b^2$ and the equation of the ellipse becomes: $$ \frac{x^2}{36+b^2}+\frac{y^2}{b^2}=1 $$ substitute $x=8.1$ and $y=4.7$ and solve the equation for $b^2$.

Answer 2

Let $d(p_1,p_2)$ mean distance between points $p_1$ and $p_2$ then we have $d(f_1,p)+d(f_2,p)=2a$,

then use the steps from here to find b.