The chord $AB$ of the parabola $y^2 = 4ax$ cuts the axis of the parabola at $C$ (C is internal to AB). If $A = (at_1^2, 2at_1)$ and $B = (at_2^2, 2at_2)$ and $AC : AB = 1 : 3$, then:
Step-by-Step Solution
Key Concept: Use section formula with the constraint AC:AB = 1:3 to find that C divides AB internally in ratio 1:2, meaning C = A + (1/3)(B-A). Since C lies on the x-axis (y = 0), the y-coordinate condition 2at₁ + (1/3)(2at₂ - 2at₁) = 0 directly yields t₂ + 2t₁ = 0. Then verify which options are consistent with this relation.
For the circle with center $C(c, 0)$ passing through points $A(at_1, 2at_1)$ and $B(at_2, 2at_2)$ on the parabola, use the condition $c = -at_1t_2$ from equidistance. The ratios $\frac{CB}{AC}$ and $\frac{AB}{AC}$ lead to $\frac{t_1 + t_2}{t_1} = 3$ and $\frac{t_1 - t_2}{t_1} = 3$. Solving these simultaneously gives $(3t_1 - 2t_2)(2t_1 + t_2) = 0$, yielding the relationship between the parameters.
Correct Answer: 2,4