Let two fixed points be $\mathrm{A}(\mathrm{ae}, 0)$ and $\mathrm{B}(-\mathrm{ae},$, $0)$. Let $\mathrm{C}(\mathrm{x}, \mathrm{y})$ be a moving point such that $\mathrm{AC}+\mathrm{CB}=$ constant $=2 \mathrm{a}$ (say)
i.e., $\sqrt{(x-a e)^{2}+(y-0)^{2}}$ $+\sqrt{(x+a e)^{2}+(y-0)^{2}}=2 a$
$\text { Or } \sqrt{x^{2}+y^{2}+a^{2} e^{2}-2 a e x}$ $+\sqrt{x^{2}+y^{2}+a^{2} e^{2}+2 a e x}=2 a$ ...(1)
Or $l+\mathrm{m}=2 \mathrm{a}$ ...(2)
Where, $l^{2}=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{a}^{2} \mathrm{e}^{2}-2 \mathrm{aex}$ ...(3)
and $m^{2}=x^{2}+y^{2}+a^{2} e^{2}+2 a e x$ ...(4)
From, (3) and (4)
$\begin{array}{l}
\quad \mathrm{m}^{2}-l^{2}=4 \mathrm{aex} \\
\text { or } \quad(\mathrm{m}-l)(l+\mathrm{m})=4 \mathrm{aex} \\
2 \mathrm{a}(\mathrm{m}-l)=4 \mathrm{aex} \quad[\text { From }(2)] \\
\mathrm{m}-l=2 \mathrm{ex} ...(5)
\end{array}$
Adding $(2)$ and $(5)$, we get
$\mathrm{m}=\mathrm{a}+\mathrm{ex}$ ...(6)
From (4) and (6),
$\begin{array}{l}
\mathrm{a}^{2}+\mathrm{e}^{2} \mathrm{x}^{2}+2 \mathrm{aex}=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{a}^{2} \mathrm{e}^{2}+2 \mathrm{aex} \\
\Rightarrow \quad \mathrm{x}^{2}\left(1-\mathrm{e}^{2}\right)+\mathrm{y}^{2}=\mathrm{a}^{2}\left(1-\mathrm{e}^{2}\right)
\end{array}$
Dividing both sides by $\mathrm{a}^{2}\left(1-\mathrm{e}^{2}\right)$, we get
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}\left(1-e^{2}\right)}=1$
Or $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, where $b^{2}=a^{2}\left(1-e^{2}\right)$
This is the equation of ellipse.