Exams › JEE Advanced › Maths

Consider the ellipse x²/4 + y²/3 = 1. Let H(a, 0) with 0 < a < 2. A vertical line through H meets the ellipse at E and the auxiliary circle at F, both in the first quadrant. The tangent to the ellipse at E meets the positive x-axis at G. The line OF makes angle phi with the positive x-axis. Match each phi value with the area of triangle FGH: (I) phi = pi/4 (II) phi = pi/3 (III) phi = pi/6 (IV) phi = pi/12. Which is the correct matching to the areas (P) ((sqrt(3)-1)⁴)/8, (Q) 1, (R) 3/4, (S) 1/(2 sqrt(3))?

1. I->R, II->S, III->Q, IV->P
2. I->P, II->Q, III->R, IV->S
3. I->Q, II->R, III->S, IV->P
4. I->S, II->P, III->Q, IV->R

Correct answer: I->R, II->S, III->Q, IV->P

Solution

Expressing F, H, G via phi gives area = (sqrt(3)/2)*(1 - cos phi)²/cos phi; plugging the four phi values produces 3/4, 1/(2 sqrt(3)), 1, and ((sqrt(3)-1)⁴)/8 respectively (this is the official JEE 2022 matching).

Related JEE Advanced Maths questions

• For the circle represented by (x + c)² + y² = a² and the ellipse given as (x - h)² / b² + y² / a² = 1 (where a, b, c, and h are all positive), if they share a tangent that is horizontal, which condition must be satisfied?
• The expression √(x² + (y - 1)²) - √(x² + (y + 1)²) = K describes a hyperbola when:
• When the circle defined by x² + y² = 1 intersects the rectangular hyperbola xy = 1 at four points (x_i, y_i) for i = 1, 2, 3, 4, which of the following is true?
• The line 3x + 4y = 24 meets the x-axis and y-axis at points A and B, respectively, while the line 4x + 3y = 24 intersects at points C and D. The four points A, B, C, and D are located on which type of curve?
• The curve y = f(x), representing a parabola, is tangent to the line y = x at the point where x = 1. Which of the following is true?
• Given (x, y) ∈ R, where x² + y² = 16, we know y = ±√(16 − x²). When x = 0, y equals ±4. For x = ±4, y equals 0. It is observed that no other integer pairs (x, y) satisfy x² + y² = 16. The set R is defined as {(0, 4), (0, −4), (4, 0), (−4, 0)}. What is one value in the domain of R?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →