Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade 11
Question:
Tangent is drawn at any fixed point $(x_1, y_1)$ on the parabola $y^2 = 4ux$. Now tangents are drawn from any point on this tangent to the circle $x^2 + y^2 = a^2$ so that all the chords of contact pass through a fixed point $(x_2, y_2)$. If $4\left(\frac{x_1}{x_2}\right) + \left(\frac{y_1}{y_2}\right)^2 = ka^2$, then $k$ equals to
Step-by-Step Solution
Key Concept: The chord of contact from an external point splits into a family of lines parametrized by $t$, all passing through a fixed point found by eliminating $t$.
For tangents at $(x_1, y_1)$ on the parabola $y^2 = 4ax$, the tangent equation is $yy_1 = 2a(x + x_1)$. The chord of contact from point $P(t, \frac{2a(t+x_1)}{y_1})$ to circle $x^2 + y^2 = a^2$ is derived by substituting the point coordinates into the chord equation. The resulting line $tx + \frac{2a(t+x_1)}{y_1}y = a^2$ can be rewritten as $(2ax_1y - a^2y_1) + t(xy_1 + 2ay) = 0$. For this to pass through the intersection of two fixed lines, the point $(x_2, y_2) = (\frac{ay_1}{2x_1}, \frac{-a^2}{x_1})$ must satisfy both the parabola condition and the constraint that $4(\frac{y_1}{x_2})(\frac{y_1}{y_2})^2 = 0$, yielding $\frac{4x_1^2}{a^2} + \frac{4x_1^2}{a^2} = 0$.
Correct Answer: 0