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Question

If α is the inclination of a tangent to the parabola y2=4ax then the distance between the tangent and a parallel normal is

A
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B
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C
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D
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Solution

The correct option is D
Let the tangent at P(at2.2at) be parallel to the normal at Q(t21,2at1)
Slope of the tangent at P is lt
Slope of the normal at Q is t1
Tangent at P is parallel to the normal at Qlt=t1t1=1t
Q=[a(lt)2,2a(lt)]=(at2,2t)
Inclination of the tangent at P is α1t=tanαt=cotα Distance between the tangent and parallel normal is
PQ=(at2at2)2+(2at22at)2=a2(t2lt2)2+4a2(t+lat)2
=a2(tlt2)2(t+lt)2+(t+lt)2
=a2(t+lt)2[(tlt)2+4]=a2(t+lt)2(t+t+lt)2=a(cotα+tanα)2=asec2α.Cosec2α

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