Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade None
Question:
$(x-1)(y-2) = 5$ and $(x-1)^2 + (y+2)^2 = r^2$ intersect at four points $A, B, C, D$ and if centroid of $\triangle ABC$ lies on line $y = 3x - 4$, then locus of $D$ is:
$y = 3x
$x^2 + y^2 + 3x + 1 = 0
$3y = x + 1
$y = 3x + 1$
Step-by-Step Solution
Key Concept: The centroid of four intersection points must satisfy the constraint equation that governs the intersection locus.
For four intersection points $(x_i, y_i)$, the centroid coordinates satisfy $\frac{\sum x_i}{4} = \frac{1 + 1}{2} = 1$ and $\frac{\sum y_i}{4} = 0$. Computing $\sum x_i = 4 - x_4$ and $\sum y_i = y_4$, the centroid lies on $y = 3x - 4$, yielding $y_4 = 3x_4$.
Correct Answer: 1