Question

For the parabola y² = 4px find the extremities of a double ordinate of length 8 p. Prove that the lines from the vertex to its extremities are at right angles. 

Answer 1

The given equation of the parabola is y² = 4px.

Let PQ be the double ordinate of length 8p of the parabola

\[y^2 = 4px\] 

Then, we have:
PR = RQ = 4p 

Let AR = x₁
Then, the coordinates of P and Q are\[\left( x_1 , 4p \right)\]   and \[\left( x_1 , - 4p \right)\] respectively.  

Now, P lies on \[y^2 = 4px\] 
∴ \[\left( 4p \right)^2 = 4p x_1\] 

\[\Rightarrow x_1 = 4p\] 

So, the coordinates of P and Q are \[\left( 4p, 4p \right)\] and \[\left( 4p, - 4p \right)\] respectively.
The coordinates of A are (0, 0). 

\[\therefore m_1 = \text{ slope of AP } = \frac{4p - 0}{4p - 0} = 1\]
\[\text{ And }, m_2 = \text{ slope of AQ }  = \frac{\left( - 4p - 0 \right)}{4p - 0} = - 1\]w

Now, 

\[m_1 m_2 = - 1\] 

Thus, AP is perpendicular to AQ.
Hence, the lines from the vertex to its extremities are at right angles.