Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade 11
Question:
Two tangents to a parabola are $x - y = 0$ and $x + y = 0$. If $(2, 3)$ is focus of the parabola, then the equation of tangent at vertex is:
4x - 6y + 5 = 0
4x - 6y + 3 = 0
4x - 6y + 1 = 0
4x - 6y + 3/2 = 0
Step-by-Step Solution
Key Concept: The latus rectum of a parabola is four times the distance from focus to the tangent at vertex.
The foot of perpendicular from focus $(2,3)$ to tangent at vertex on line $x-y=0$ is found using the perpendicularity condition. Computing the foot on $x-y=0$ gives $\left(\frac{5}{2}, \frac{5}{2}\right)$, and the foot on $x+y=0$ gives $\left(-\frac{1}{2}, \frac{1}{2}\right)$. The tangent at vertex passes through these two points with equation $4x-6y+5=0$. The latus rectum is calculated as $4×\frac{|8-18+5|}{\sqrt{52}}=\frac{10}{\sqrt{13}}$, and using $\frac{1}{SP}+\frac{1}{SQ}=\frac{2\sqrt{13}}{5}$.
Correct Answer: 1