Question

AB,AC are tangents to a parabola y2=4ax.p1,p2 & p3 are the length of the perpendiculars from A,B & C respectively on any tangent to the curve, then p2,p1,p3 are in:

A
A.P.
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B
G.P
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C
H.P
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D
none of these
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Solution

The correct option is B G.Py2=4axLet B(at21,2at1);c=(at22,2at2)So A=(at1t2,a(t1+t2))Eg of tangent: y=mx+amLength of perpendicular from BP2=∣∣ ∣ ∣∣2at1−mat21−am√1+m2∣∣ ∣ ∣∣=∣∣am∣∣(mt1−1)2√1+m2Length of perpendicular from C.P3=∣∣ ∣ ∣∣2at2−mat22−am√1+m2∣∣ ∣ ∣∣=∣∣am∣∣(mt2−1)2√1+m2Length of perpendicular from AP1=∣∣ ∣ ∣∣mat1t2−a(t1+t2)+am√1+m2∣∣ ∣ ∣∣=∣∣am∣∣(mt1−1)(mt2−1)√1+m2P21=P2P3So P2,P1,P3 are in G.P.

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