Question

For the parabola $y^2=4px$ find the extremities of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.

Answer 1

Let PQ be the double ordinate of length 8p of the parabola $y^2=4px$

Then,PR=QR=4p

let AR=$x_1\text{Then,the coordinates of P and Q are}(4p,4p)and(4p,-4p)respectively.\text{Since P lies on}{y}^2=4px\therefore (4p{)}^2=4p{x}_1\Rightarrow x_1=4p\text{So,coordinates of P and Q are (4p,4p) and (4p,-4p) respectively.}\Rightarrow \text{The extremities of a double ordinate are 4p,4p) and (4p,-4p)}\text{Also,the coordinates of the vertex A are (0,0)}\therefore m_1=slopeofAP=\frac{4p-0}{4p-0}=1and,m_2=slopeofAQ=\frac{-4p-0}{4p-0}Clearly,m_1{m}_2=-1Hence,AP\perp AQ\therefore \text{The lines from the vertex to its extremities are at right angles.}$