Tangent are drawn to the ellipse $3{\mathrm{x}}^2+5{\mathrm{y}}^2=32\text{ and }25{\mathrm{x}}^2+9{\mathrm{y}}^2=450$ passing through the point (3,5). The number of such tangents are …..

Putting (3, 5) in the equation of the ellipse, we get S₁= +ive and S₂ = 0 so that the point (3, 5) lies outside S₁ = 0 and hence two tangents can be drawn through the point (3, 5). The point (3, 5) lies on S₂ = 0 and only one tangent can be drawn. Thus the total no. of tangents passing through (3, 5) to the two ellipses is 2 +1= 3 (B) (C) The equation of two perpendicular chords drawn through each of the foci be $\mathrm{y}=\mathrm{m}(\mathrm{x}-\mathrm{ae})$ and 
$\mathrm{y}=\frac{1}{\mathrm{m}}(\mathrm{x}+\mathrm{ae})$ Locus of their point of intersection P is obtained by eliminating the variable m. Multiplying the equations of chords, we have $\mathrm{y}=-\left({\mathrm{x}}^2-{\mathrm{a}}^2{\mathrm{e}}^2\right)\text{ or }{\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2-{\mathrm{b}}^2$
Above represents the director circle of hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ which we know is the locus of the point of intersection of perpendicular tangents QP and QR.
Hence $\mathrm{\angle }\mathrm{QPR}=\mathrm{\pi }/2$