Question

Find the equation of the ellipse having centre at $(1,2)$, one focus at $(6,2)$ and passing through the point $(4,6)$.

Answer 1

Let the equation of the ellipse be

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

Distance between the focus and centre

$⟹\sqrt{(6-1{)}^2+(2-2{)}^2}=\sqrt{a^2-b^2}$

$⟹25=a^2-b^2$

$⟹a^2=25+b^2$

As the ellipse passes through the point $(4,6)$ we have,

$\frac{(4-1{)}^2}{a^2}+\frac{(6-2{)}^2}{b^2}=1$

$9b^2+16a^2=a^2{b}^2$

$9b^2+16(25+b^2)=(25+b^2)b^2$

$b^4=16\ast 25$

$⟹b^2=20$

$⟹a^2=45$