AB,AC are tangents to a parabola y2=4ax, if l1,l2,l3 are the lengths of perpendiculars from A,B,C on any tangents to the parabola, then
A
l1,l2,l3 are in GP
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B
l2,l1,l3 are in GP
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C
l3,l1,l2 are in GP
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D
l3,l2,l1 are in GP
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Solution
The correct option is Al2,l1,l3 are in GP Let the coordinates of B and C be (at21,2at1) and (at22,2at2) respectively.
Then, the coordinates of A are (at1t2,a(t1+t2)). The equation of any tangent to y2=4ax is ty=x+at2. l1=at1t2−a(t1+t2)t+at2√1+t2 l2=at21−2at1+at2√1+t2 and l3=at22−2at2+at2√1+t2 Clearly, l2l3=l21 Therefore, l2,l1,l3 are in GP.