Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
Tangent is drawn at any point $(x_1, y_1)$ other than vertex on the parabola $y^2 = 4ax$. If tangents are drawn from any point on this tangent to the circle $x^2 + y^2 = a^2$ such that all the chords of contact pass through a fixed point $(x_2, y_2)$, then :
$x_1, a, x_2$ are in G.P.
$\frac{y_1}{2}, a, y_2$ are in G.P.
$-4, \frac{y_1}{x_1}, \frac{x_1}{x_2}$ are in G.P.
$x_1 x_2 + y_1 y_2 = a^2$
Step-by-Step Solution
Key Concept: The chord of contact traces a family of lines through the locus of intersection of two fixed lines as the parameter $t$ varies.
For a point $(x_1, y_1) = (at^2, 2at)$ on the parabola $y^2 = 4ax$, the tangent is $ty = x + at^2$. The chord of contact with respect to circle $x^2 + y^2 = a^2$ is derived by substituting the point into the circle equation, yielding a family of lines passing through the fixed point of intersection of $ty - a = 0$ and $x + \frac{y}{t} = 0$. This fixed point is $\left(-\frac{a}{t^2}, \frac{a}{t}\right)$. From the conditions $x_1x_2 = -a^2$, $y_1y_2 = 2a^2$, and the relationships $\frac{x_1}{x_2} = -t^4$ and $\frac{y_1}{y_2} = 2t^2$, we derive $4\frac{x_1}{x_2} + \left(\frac{y_1}{y_2}\right)^2 = 0$.
Correct Answer: 2,3,4