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show that if the normals at the points P(at 1 ,2at 1 ) and G(at 2 2 ,2at 2 ) to the parabola y 2 =4ax, intersect on the paarabola, then t 1 t 2 = 2.
Let the normals at P and Q meet at R(at2, 2at).Then t = – t1 – 2/t1 and t = – t2 – 2/t2 .Therefore t1 + 2/t1 = t2 + 2/t2 ⇒ (t1 – t2) = 2(t1-t2 )/(t1 t2 ) ⇒ t1t2 = 2.
Let the normals at P and Q meet at R(at2, 2at).
Then t = – t1 – 2/t1 and t = – t2 – 2/t2 .
Therefore t1 + 2/t1 = t2 + 2/t2 ⇒ (t1 – t2) = 2(t1-t2 )/(t1 t2 ) ⇒ t1t2 = 2.
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