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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.
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Solution

The given equation of the parabola is y2 = 4px.
Let PQ be the double ordinate of length 8p of the parabola
PR = RQ = 4p
Then, the coordinates of P and Q are\[\left( x_1 , 4p \right)\] and \[\left( x_1 , - 4p \right)\] respectively.
∴ \[\left( 4p \right)^2 = 4p x_1\]
The coordinates of A are (0, 0).
\[\text{ And }, m_2 = \text{ slope of AQ } = \frac{\left( - 4p - 0 \right)}{4p - 0} = - 1\]w
Hence, the lines from the vertex to its extremities are at right angles.
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