Question

An ellipse has its center at $(1,-1)$ and semi- major axis $=$ $8$, which passed through the point $(1,3)$. If the major axis is parallel to $x$-axis then the equation of ellipse is

Answer 1

Equation of ellipse, with axes as its major and minor axes

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

if center is at $(1,-1)$

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

semi- major axis $=a=8$

$\Rightarrow$ Equation becomes.

$\frac{(x-1{)}^2}{64}+\frac{(y+1{)}^2}{b^2}=1$

The ellipse parts through (1,3)

$\Rightarrow \frac{(1-1{)}^2}{64}+\frac{(3+1{)}^2}{b^2}=1$

$\Rightarrow 0+\frac{16}{b^2}=1$

$\Rightarrow b^2=16$

$\Rightarrow$ Equation of ellipse is

$\frac{(x-1{)}^2}{64}+\frac{(y+1{)}^2}{16}=1$