The tangent at any point P on the ellipse meets the tangent at the vertices $\mathrm{A} \mathrm{\&} {\mathrm{A}}^1$ of the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ at L and M respectively .Then$\left(AL\right)\left({\mathrm{A}}^1\mathrm{M}\right)=$

Equation of tangent to the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ at  $P\left(\theta \right)$ is'$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$

Equation  of tangents at the vertices $\mathrm{A} \mathrm{\&} {\mathrm{A}}^1$  are $x=\pm a$

$$\begin{gathered} L\left(a, b\left(\frac{1-\cos \theta }{\sin \theta }\right)\right)L^{\text{'}}\left(-a, b\left(\frac{1+\cos \theta }{\sin \theta }\right)\right) \\ \Rightarrow AL.A^{\text{'}}M=b\left(\frac{1-\cos \theta }{\sin \theta }\right) b\left(\frac{1+\cos \theta }{\sin \theta }\right)=b^2 \end{gathered}$$