Given : Equations of ellipses
$3x^2 + 5y^2 =32 \, \, \, \, \, \, \, \, \, \, \, $..(1)
& $25x^2 + 9y^2=450 \, \, \, \, \, \, \, \, \, \, \, $...(2)
Tangents to the ellipse (1) & (2) are passing through the point (3, 5)
$\therefore \, \, 3(3)^2 + 5(5)^2-32 = 27+75-32>0$
So the given point lies outsides the ellipse. Hence, two real tangents can be drawn from the point to the ellipse
& $25(3)^2 + 9(5)^2-450=225+225-450=0$
$\therefore $ The point lie on the ellipse. Hence one
real tangent can be drawn.
$\therefore \, \, $ No. of real tangents = 3