Question

Find the equation to the ellipses, whose centres are the origin, whose axes are the axes of coordinates, and which pass through $\left(\alpha \right)$ the points $(2,2)$, and $(3,1)$ and $\left(\beta \right)$ the points $(1,4)$ and $(-6,1)$.

Answer 1

We will have general equation of ellipse for centre at $(0,0)$

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

Putting $(2,2),(3,1)$

$\frac{4}{a^2}+\frac{4}{b^2}=1$

$\frac{9}{a^2}+\frac{1}{b^2}=1$

We have,

$\frac{1}{a^2}=\frac{3}{32},\frac{1}{b^2}=\frac{5}{32}$

We get,

$3x^2+5y^2=32$

For second part,

Putting $(1,4),(-6,1)$

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

$\frac{1}{a^2}+\frac{16}{b^2}=1$

$\frac{36}{a^2}+\frac{1}{b^2}=1$

We have

$\frac{1}{a^2}=\frac{3}{115},\frac{1}{b^2}=\frac{7}{115}$

We get,

$3x^2+7y^2=115$