Foci of an ellipse are at $\mathrm{S}(1,7), {\mathrm{S}}^{\text{'}} (1, -3)$ . The point P is on the ellipse such that $\mathrm{SP}=7, {\mathrm{S}}^{\text{'}}\mathrm{P}=5$. Then the equation of the ellipse is

$$\begin{gathered} \mathrm{S}(1, 7) {\mathrm{S}}^1(1, -3) \\ \mathrm{centre}=(1, 2)=(h,k) \\ major axis is parallel to y-axis \\ So \mathrm{SP}+{\mathrm{S}}^1\mathrm{P}=2\mathrm{b}=12 \\ \Rightarrow \mathrm{b}=6 \\ and \mathrm{SS}\text{'}=2\mathrm{be}=10 \\ \Rightarrow be=5 \\ \Rightarrow {\mathrm{b}}^2{\mathrm{e}}^2=25 \\ Now {\mathrm{b}}^2-{\mathrm{a}}^2= {\mathrm{b}}^2{\mathrm{e}}^2=25 \\ \Rightarrow {\mathrm{a}}^2=11 \\ \therefore \mathrm{eq}uation of ellipse is \frac{{\left(\mathrm{x}-1\right)}^2}{11}+\frac{{\left(\mathrm{y}-2\right)}^2}{36}=1 \end{gathered}$$