Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade 11
Question:
If equation of tangent at P, Q and vertex A of a parabola are $3x + 4y - 7 = 0$, $2x + 3y - 10 = 0$ and $x - y = 0$ respectively, then:
Focus is (4, 5)
Length of latus rectum is $2\sqrt{2}$
Axis is $x + y - 9 = 0$
Vertex is $\left(\frac{9}{2}, \frac{9}{2}\right)$
Step-by-Step Solution
Key Concept: The directrix and tangent at vertex of a hyperbola, along with focal points, determine the latus rectum and vertex position through geometric relationships.
For hyperbola with $R(1,1)$ and $T(2,2)$: the right directrix (RS) has equation $4x - 3y - 1 = 0$ and the tangent at vertex (TS) has equation $3x - 2y - 2 = 0$. The latus rectum length is $4 \times \frac{1}{\sqrt{2}} = 2\sqrt{2}$. The vertex is at $(\frac{9}{2}, \frac{9}{2})$ where the axis $x + y - 9 = 0$ intersects.
Correct Answer: 1,2,3,4