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Question

For the parabola y2=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.

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Solution

Let PQ be the double ordinate of length 8p of the parabola y2=4px

Then,PR=QR=4p

let AR=x1 Then,the coordinates of P and Q are(4p,4p) and (4p,4p) respectively.Since P lies on y2=4px (4p)2=4px1 x1=4pSo,coordinates of P and Q are (4p,4p) and (4p,-4p) respectively. The extremities of a double ordinate are 4p,4p) and (4p,-4p) Also,the coordinates of the vertex A are (0,0) m1=slope of AP =4p04p0 =1 and,m2=slope of AQ =4p04p0 Clearly,m1m2=1 Hence,APAQ The lines from the vertex to its extremities are at right angles.


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