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Conic Sections · Question 8

JEE Advanced 2024 Paper 1MCQMedium📈 Topic in 33% papers+4−1
1

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

Answer

(1) q=2, p=3 3

Solution

S is in the first quadrant, so the tangent at an end of the minor axis is y=2, giving q=2.

R=(3,0); area ORT= 12 3 |yT|= 32 yT=-1

T=(3 ,2 ) with =- 12, = 32, i.e. T= ( 3 32,-1 )

Tangent at T: x3 32- y4=1

Put y=2: 3x6= 32 x=3 3=p

Answer: (A)

JEE Advanced 2024 Paper 1

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2

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)
JEE Advanced 2024 Paper 1
3

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)
JEE Advanced 2024 Paper 2
4

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 .

JEE Advanced 2024 Paper 2

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