Question

Show that the point (0,3+√5) lies on the ellipse with foci (2,3) and (-2,3) and semi major axis 3.

Answer 1

The foci are (2,3) and (-2,3), so obviously (0,3) is a center and c=2. Semi major axis a=3. Therefore we can find a semi minor axis b from the formula $b^2+c^2=a^2,b=\sqrt{a^2-c^2}=\sqrt{9-4}=\sqrt{5}$ . We put these values in the general equation of the ellipse and we get:

$\frac{x^2}{9}+\frac{(y-3)^2}{5}=1$

Let's put $(0,3+\sqrt{5})$ in this equation:

$\frac{0}{9}+\frac{(3+\sqrt{5}-3)^2}{5}=\frac{(\sqrt{5})^2}{5}=1$ , which proves that this point lies on the indicated ellipse.