Exams › JEE Main › 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 Main Maths questions

• For the pair of parallel straight lines represented by 9x² - 6xy + y² + 18x - 6y + 8 = 0, what is the separation between them?
• An ellipse has its two foci 10 units apart, and the length of its latus rectum is 15. If its axes are taken as the coordinate axes, which equation represents the ellipse?
• On the segment joining A(0, 0) and B(3a, 0), choose points P and Q so that AP = PQ = QB. Three circles are then constructed with AP, PQ, and QB as their respective diameters. If a point S is such that the sum of the squares of the tangents drawn from S to these three circles is b², then the locus of S is
• Find the coordinates of the midpoint of the chord cut by the circle x² + y² + 4x - 2y - 3 = 0 on the line y = x + 2.
• A hyperbola has a transverse axis of length 2 sin θ and is confocal with the ellipse 3x² + 4y² = 12. Its equation is
• Three points E, F and G are chosen on the parabola y² = 4ax such that their y-coordinates form a geometric progression. The point where the tangents at E and G meet lies on the

⚔️ Practice JEE Main Maths free + battle 1v1 →