Conic Sections · Question 6

Let P(x1,y1) and Q(x2,y2) be two distinct points on the ellipse x29+ y24=1 such that y1>0, and y2>0. Let C denote the circle x2+y2=9, and M be the point (3,0). Suppose the line x=x1 intersects C at R, and the line x=x2 intersects C at S, such that the y-coordinates of R and S are positive. Let ROM= 6 and SOM= 3, where O denotes the origin (0,0). Let |XY| denote the length of the line segment XY. Then which of the following statements is (are) TRUE?

1. The equation of the line joining P and Q is 2x+3y=3(1+ 3)
2. The equation of the line joining P and Q is 2x+y=3(1+ 3)
3. If N2=(x2,0), then 3|N2Q|=2|N2S|
4. If N1=(x1,0), then 9|N1P|=4|N1R|

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

Let the straight line y=2x touch a circle with center (0, ), >0, and radius r at a point A1. Let B1 be the point on the circle such that the line segment A1B1 is a diameter of the circle. Let +r=5+ 5. Match each entry in List-I to the correct entry in List-II.

1. (P) → (4) (Q) → (2) (R) → (1) (S) → (3)
2. (P) → (2) (Q) → (4) (R) → (1) (S) → (3)
3. (P) → (4) (Q) → (2) (R) → (5) (S) → (3)
4. (P) → (2) (Q) → (4) (R) → (3) (S) → (5)

Let A1,B1,C1 be three points in the xy-plane. Suppose that the lines A1C1 and B1C1 are tangents to the curve y2=8x at A1 and B1, respectively. If O=(0,0) and C1=(-4,0), then which of the following statements is (are) TRUE?

1. The length of the line segment OA1 is 4 3
2. The length of the line segment A1B1 is 16
3. The orthocenter of the triangle A1B1C1 is (0,0)
4. The orthocenter of the triangle A1B1C1 is (1,0)

A normal with slope 1 6 is drawn from the point (0,- ) to the parabola x2=-4ay, where a>0. Let L be the line passing through (0,- ) and parallel to the directrix of the parabola. Suppose that L intersects the parabola at two points A and B. Let r denote the length of the latus rectum and s denote the square of the length of the line segment AB. If r:s=1:16, then the value of 24a is .