The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is

For the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$,match the lines given in List-$I$ with their equations given in List-$II$.

List-$I$	List-$II$
$(P)$ Directrix corresponding to the focus $(-3, 0)$	$(1)$ $y = 4$
$(Q)$ Tangent at the vertex $(0, 4)$	$(2)$ $3x = 25$
$(R)$ Latus rectum through $(3, 0)$	$(3)$ $x = 3$
	$(4)$ $y + 4 = 0$
	$(5)$ $x + 3 = 0$
	$(6)$ $3x + 25 = 0$

The tangents drawn from the point $P(3, 4)$ to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ touch the ellipse at points $A$ and $B$. The equation of the locus of a point which is equidistant from point $P$ and the line $AB$ is:

The eccentricity of the ellipse $4x^2 + 25y^2 = 100$ is

Find the equation for the ellipse that satisfies the given conditions: Foci $(\pm 3, 0)$,$a = 4$.

If the line $2x - 3y + 4 = 0$ cuts the ellipse $x = 3 \cos \theta, y = 5 \sin \theta$ at points $A$ and $B$,and $(\alpha, \beta)$ is the midpoint of $\overline{AB}$,then $3\beta - 2\alpha =$