Question

Derive equation of the ellipse in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Answer 1

Consider the co-ordinate system without any shifting or translation of axes and origin,

Consider the length of major axis be $2a$,

We know that in an ellipse $d_1+d_2=2a$,

$\sqrt{{(x+c)}^2+y^2}+\sqrt{{(x-c)}^2+y^2}=2a⟹\sqrt{{(x+c)}^2+y^2}=2a-\sqrt{{(x-c)}^2+y^2}$

Squaring on both sides,

${(x+c)}^2+y^2=4a^2-4a\sqrt{{(x-c)}^2+y^2}+x^2-2xc+y^2$

Rearranging and squaring the terms gives,

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