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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.P
y2=4ax
Let B(at21,2at1);c=(at22,2at2)
So A=(at1t2,a(t1+t2))
Eg of tangent: y=mx+am
Length of perpendicular from B
P2=∣ ∣ ∣2at1mat21am1+m2∣ ∣ ∣=am(mt11)21+m2
Length of perpendicular from C.
P3=∣ ∣ ∣2at2mat22am1+m2∣ ∣ ∣=am(mt21)21+m2
Length of perpendicular from A
P1=∣ ∣ ∣mat1t2a(t1+t2)+am1+m2∣ ∣ ∣=am(mt11)(mt21)1+m2
P21=P2P3
So P2,P1,P3 are in G.P.

1194185_1265911_ans_6250964857a449edaabc751cb939f4bf.jpg

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