Parabola Question - None

Question

The chord AC of the parabola $y^2 = 4ax$ subtends an angle of $90°$ at points B and D on the parabola. If points $A, B, C$ and $D$ are represented by $(at_i^2, 2at_i)$, $i = 1, 2, 3, 4$ respectively, then find the value of $\frac{t_2 + t_4}{t_1 + t_3}$.

Accepted Answer

Step 1: Parametric points on parabola $y^2 = 4ax$ are $(at^2, 2at)$.

Let $A = (at_1^2, 2at_1)$, $B = (at_2^2, 2at_2)$, $C = (at_3^2, 2at_3)$, $D = (at_4^2, 2at_4)$.

Step 2: For angle $\angle ABC = 90°$, vectors $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are perpendicular:

$\overrightarrow{BA} \cdot \overrightarrow{BC} = 0$

Step 3: The condition that AC subtends $90°$ at both B and D gives:

$t_1 t_3 = -4$ (from $\angle ABC = 90°$)

$t_1 t_3 = -4$ (from $\angle ADC = 90°$)

Step 4: Using the property that if two chords subtend $90°$ at two points on the parabola:

$t_2 + t_4 = t_1 + t_3$

Therefore, $\frac{t_2 + t_4}{t_1 + t_3} = \boxed{1}$.

Question:

Step-by-Step Solution

Master Parabola with Mathbee