Exams › JEE Advanced › Maths

Consider a family of ellipses, all centred with axes along the coordinate axes and all sharing the same major axis length 2a, but with varying minor axis. The tangents drawn at the ends of the latus rectum of these ellipses all pass through certain fixed points. Identify those fixed points.

1. (0, a) and (0, -a)
2. (a, 0) and (-a, 0)
3. (0, 0) only
4. (a, a) and (-a, -a)

Correct answer: (0, a) and (0, -a)

Solution

The tangent at a latus-rectum end simplifies to (e x + y)/a = 1; using b² = a²(1-e²) one finds it always passes through (0, a) (and by symmetry (0, -a)).

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 →