Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
If $A(3, 0)$ and $B(6, 0)$ are two fixed points and $U(a,b)$ is a variable point in the plane. $AU$ and $BU$ meet the y-axis at $C$ and $D$ respectively and $AD$ meet $OU$ at $V$. Then the coordinate of the point through which $CV$ always passes is:
(3, 4)
(4, 0)
(0, 2)
(2, 0)
Step-by-Step Solution
Key Concept: A family of lines passes through a fixed point when the equation can be rewritten to eliminate parameters.
The equation of the chord of contact (or related conic property) from a point is $y = \frac{3\beta}{(3-\alpha)}\left(1-\frac{x}{2}\right)$. This simplifies to show that the line passes through the fixed point $(2, 0)$, independent of the parameters $\alpha$ and $\beta$.
Correct Answer: 4