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Question

A pair of tangents are drawn which are equally inclined to a straight line whose inclination to the axis is α; prove that the locus of their point of intersection is the straight line
y=(xa)tan2α

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Solution

Let the point of intersection be P(h,k)

Equation of tangent to y2=4ax in slope form is

y=mx+ammy=m2x+am2xmy+a=0

It passes through P(h,k)

m2hmk+a=0

The equation is quadractic in m

m1+m2=kh=kh.....(i)m1m2=ah.........(ii)

Slope of line which makes equal angle with the tangents =tanα

m1tanα1+m1tanα=tanαm21+m2tanαm1+m1m2tanαtanαm2tan2α=tanαm2+m1tan2αm1m2tanαm1+m2(m1+m2)tan2α2tanα+2m1m2tanα=0(1tan2α)(m1+m2)2tanα+2tanαm1m2=0(m1+m2)2tanα1tan2α+2tanα1tan2αm1m2=0(m1+m2)tan2α+tan2αm1m2=0

using (i) and (ii)

khtan2α+tan2αah=0khtan2α+atan2α=0k=htan2αatan2α

Replacing h by x and k by y

y=tan2α(xa)



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