Conic Sections

Let P be the point in the first quadrant where the given ellipses intersect. If is the acute angle between the tangents to the given ellipses at the point P, then the value of 4 is .

Consider the ellipse x29+ y24=1. Let S(p,q) be a point in the first quadrant such that p29+ q24>1. Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive x-coordinate and O be the center of the ellipse. If the area of the triangle ORT is 32, then which of the following options is correct?

1. q=2, p=3 3
2. q=2, p=4 3
3. q=1, p=5 3
4. q=1, p=6 3