Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade 11
Question:
$$\frac{x^2}{r^2 - r - 6} + \frac{y^2}{r^2 - 6r + 5} = 1$$ will represents the ellipse, if $r$ lies in the interval
(-∞, -2)
(3, ∞)
(5, ∞)
(1, ∞)
Step-by-Step Solution
Key Concept: Intersection conditions for a circle and parabola require both discriminant inequalities to hold simultaneously.
For the circle $(x - r)^2 + y^2 = r^2$ to intersect the parabola $y^2 = 4x$, we need $r^2 - r - 6 > 0$ and $r^2 - 6r + 5 > 0$. Factoring gives $(r - 3)(r + 2) > 0$ and $(r - 1)(r - 5) > 0$. The solution is $r 5$.
Correct Answer: 1,3