If P(x₁, y₁) is any point on the Ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1(\mathrm{a}>\mathrm{b})$ whose foci are ‘S’ and ‘S¹’ then $SP+S^{\mathrm{\prime }}P=2\mathrm{a}\left(\text{ constant }\right)$

The equation of the ellipse is $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1(\mathrm{a}>\mathrm{b})$
Let S, S' be the foci and ZM, Z'M' be the directrices. P(x, y) is a point on the ellipse. Draw PL perpendicular to x-axis and $\overleftrightarrow{\mathrm{M}\text{'}\mathrm{M}}$ through P perpendicular to both the two directrices (See Fig 4.1). By the definition of the ellipse
$\mathrm{SP}=\mathrm{ePM}=\mathrm{e}\left(\mathrm{LZ}\right)$
$\begin{array}{l}\therefore \mathrm{SP}=\mathrm{e}(\mathrm{CZ}-\mathrm{CL})=\mathrm{e}\left(\frac{\mathrm{a}}{\mathrm{e}}-\mathrm{x}\right)=\mathrm{a}-\mathrm{ex}\\ \begin{array}{ll}{\mathrm{S}}^{\mathrm{\prime }}\mathrm{P}={\mathrm{ePM}}^{\mathrm{\prime }}& =\mathrm{e}\left({\mathrm{LZ}}^{\mathrm{\prime }}\right)=\mathrm{e}\left(\mathrm{CL}+{\mathrm{CZ}}^{\mathrm{\prime }}\right)\\ & =\mathrm{e}\left(\mathrm{x}+\frac{\mathrm{a}}{\mathrm{e}}\right)=\mathrm{a}+\mathrm{ex}\\ \therefore \mathrm{SP}+{\mathrm{S}}^{\mathrm{\prime }}\mathrm{P}& =\mathrm{a}-\mathrm{xe}+\mathrm{a}+\mathrm{xe}=2\mathrm{a}=\left(\text{ constant }\right)\\ & \end{array}\end{array}$