The chord $AC$ of the parabola $y^2 = 4ax$ subtends an angle of $90°$ at points $B$ and $D$ on the parabola. If $A, B, C$ and $D$ are represented by $t_1, t_2, t_3$ & $t_4$, then find the value of $\left|\frac{t_2 + t_4}{t_1 + t_3}\right|$.
Step-by-Step Solution
Key Concept: For a parabola y² = 4ax with parametric points (at², 2at), the condition that chord AC subtends 90° at point B requires the dot product of vectors BA and BC to equal zero. This yields the constraint (t₁+t₂)(t₂+t₃) = -4. Applying this same constraint for point D gives (t₁+t₄)(t₄+t₃) = -4, which forces t₂+t₄ = -(t₁+t₃).
Four points $A(at_1^2, 2at_1)$, $B(at_2^2, 2at_2)$, $C(at_3^2, 2at_3)$, $D(at_4^2, 2at_4)$ on parabola $y^2 = 4ax$ satisfy conditions $\angle ABC = 90°$ and $\angle ADC = 90°$. These constraints give $(t_1+t_2)(t_2+t_3) = -4$ and $(t_1+t_4)(t_3+t_4) = -4$, leading to $t_2 + t_4 = -(t_1+t_3)$.
Correct Answer: 1