If the normals at four points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ and $(x_4,y_4)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are concurrent then $\left(\sum _{i=1}^4\cos {\alpha }_i\right)\left(\sum _{i=1}^4\sec {\alpha }_i\right)$ is (where ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ are the eccentric angles of the given points)

Equation of normal at a point P (x, y) is $\frac{a^{2}x}{x_1}-\frac{b^{2}y}{y_1}=a^2-b^2$
If this normal passes through a fixed point (h, k)
$\frac{a^{2}h}{x_1}-\frac{b^{2}k}{y_1}=a^2-b^2\Rightarrow \frac{b^{2}k}{y_1}=\frac{a^{2}h}{x_1}-(a^2-b^2)$ 
$P(x_1,y_1)$ is a point on the ellipse
$\Rightarrow \frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=1\Rightarrow b^2{x}_1^2+a^2{y}_1^2=a^2{b}^2$ 
$\Rightarrow b^2{x}_1^2+a^2{\left(\frac{b^{2}k}{\frac{a^{2}h}{x_1}-(a^2-b^2)}\right)}^2=a^2{b}^2$ 
$\Rightarrow {(a^2-b^2)}^2{x}_1^4-2h{a}^2(a^2-b^2)x_1^3-x_1^2+2a^{4}h(a^2-b^2)x_1-a^6{h}^2=0$ 
$x_1,x_2,x_3,x_4$ are the roots of this equation $\Rightarrow \sum x_i\times \sum \frac{1}{x_i}=4$