For an ellipse$\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$with vertices A and A' tangent drawn at the point P in the first quadrant meets the y-axis at Q and the chord A' P meets y-axis in M.O is the origin then OQ² – MQ^{2 .}

$$\begin{gathered} Given ellipse \frac{x^2}{9}+\frac{y^2}{4}=1 \\ where a^2=9,b^2=4 \\ and vertices are A=(3,0) and A\text{'}=(-3,0) \\ Let P=(3\cos \theta ,2\sin \theta ) be any point on ellipse \end{gathered}$$

Now  tangent at P is $\frac{\mathrm{x}}{3}\cos \mathrm{\theta }+\frac{\mathrm{y}}{2}\sin \mathrm{\theta }=1$

$$\begin{gathered} It meets y-axis at Q(0,2\cos \mathrm{ec}\mathrm{\theta }) \\ Now equation os A\text{'}P is y=\frac{2\sin \theta }{3(\cos \theta +1)}(x+3) \\ It meets y-axis at \mathrm{M}\left(0,\frac{2\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}\right) \end{gathered}$$
 

$\Rightarrow {\mathrm{OQ}}^2-{\mathrm{MQ}}^2=4$