Conic Sections · Question 3

Consider the ellipse E given by x218+ y212=1. Let H be the hyperbola whose eccentricity is the reciprocal of the eccentricity of E and whose foci are the same as that of E. Let P and Q be the points of intersection of H and the parabola 5 y=x2 in the first quadrant. Let d be the distance between P and Q. If a and b are the integers such that d2=a+b 5, then the value of a-b is .

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 .

Let S denote the locus of the point of intersection of the pair of lines 4x-3y=12 , 4 x+3 y=12, where varies over the set of non-zero real numbers. Let T be the tangent to S passing through the points (p,0) and (0,q), q>0, and parallel to the line 4x- 3 2y=0. Then the value of pq is

1. -6 2
2. -3 2
3. -9 2
4. -12 2

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 .