Exams › JEE Advanced › Maths

Define ellipses {E1, E2,...} and rectangles {R1, R2,...} as follows. E1: x²/9 + y²/4 = 1. R1 is the largest-area rectangle (sides parallel to the axes) inscribed in E1. For n > 1, En is the largest-area ellipse inscribed in R_(n-1), and Rn is the largest-area rectangle (axis-parallel sides) inscribed in En. Which of the following is/are correct?

1. The eccentricities of E18 and E19 are equal, the latus rectum of E9 is 1/6, and the sum of areas of Rₙ is less than 24 for every N
2. The eccentricities of E18 and E19 are NOT equal
3. The distance of a focus from the centre in E9 is sqrt(5)/32
4. The length of latus rectum of E9 is 1/3

Correct answer: The eccentricities of E18 and E19 are equal, the latus rectum of E9 is 1/6, and the sum of areas of Rₙ is less than 24 for every N

Solution

Since semi-axes scale uniformly, all En have the same eccentricity (so E18 and E19 are equal); latus rectum of E9 is 1/6 and the rectangle areas sum to a convergent series bounded above by 24.

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 →