Find an equation for the ellipse with foci $(\pm 4,0)$ passing through $(-4,1.8)$

Question

Find an equation for the conic that satisfies the given conditions:

Ellipse, Foci $(-4,0)$ and $(4,0)$, passes through $(-4,1.8)$.

I know how to do these questions with the vertices, but I'm kinda lost figuring this one out.

Answer 1

Let $c = \sqrt{a^2-b^2}$. We know $c = 4$; so $16=a^2-b^2$ and $b^2+16=a^2$. Now you substitute in: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

You obtain: $$\frac{4^2}{b^2+16}+\frac{1.8^2}{b^2}=1$$

Solving for $b$, we find $b=3$ and $a=5$. The equation is: $$\frac{x^2}{25}+\frac{y^2}{9}=1$$

Answer 2

Hint: One way to define an ellipse is that the sum of the distances from a point on the ellipse to the two foci is the same for all points of the ellipse. Can you come up with an equation that expresses this fact?

Answer 3

Matteo (https://math.stackexchange.com/users/686644/matteo), Find an equation for the ellipse with foci $(\pm 4,0)$ passing through $(-4,1.8)$, URL (version: 2019-07-23): https://math.stackexchange.com/q/3301684