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Question

If the straight line y=mx+c touches the hyperbola x2a2y2b2=1 then prove that c2=a2m2b2

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Solution

It is given that the line y=mx+c touches the hyperbola.
Consider the tangent of hyperbola in parametric form
xsecθaytanθb=1
ytanθb=xsecθa1
y=bxsecθatanθbtanθ
y=basinθxbtanθ
On comparing with y=mx+c, we get
m=basinθ and c=btanθ
a2m2b2=a2(basinθ)2b2
a2m2b2=b2sin2θb2
a2m2b2=b2(1sin2θ)sin2θ
a2m2b2=b2cos2θsin2θ
a2m2b2=b2tan2θ
a2m2b2=c2
Hence proved.

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