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Question

The two parabolas y2=4a(xl) and x2=4a(yl) always touch one another, the quantities l and l' being both variable; prove that the locus of their point of contact is the curve xy=4a2.

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Solution

y2=4a(xl).....(i)x2=4a(yl).....(ii)
Let the point of contact be P(h,k)
Both the parabolas will have a common tangent at their point of contact
Let slope tangent at P to (i) be m1
2ydydx=4am1=(dydx)(h,k)=2ay=2ak
Let slope tangent at P to (ii) be m2
2x=4adydxm2=(dydx)(h,k)=x2a=h2a
Now m1=m2
2ak=h2ahk=4a2
Generalising the equation
xy=4a2
Hence proved.

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