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:
Step-by-Step Solution
Key Concept: For a parabola, the tangent at vertex is perpendicular to the axis of symmetry. The axis passes through the focus and is perpendicular to the tangent at vertex. By finding perpendiculars from focus (2,3) to the two given tangents, their feet determine a chord of contact whose perpendicular bisector through the focus gives the axis direction, enabling derivation of 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