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Question

A normal to the hyperbola x24y21=1 has equal intercepts on positive x and y axes. If this normal touches the ellipse x2a2+y2b2=1, then the value of 3(a2+b2) is

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Solution

The equation of the normal to the hyperbola x24y21=1 at (2secθ,tanθ) is 2xcosθ+ycotθ=5 ...(i)

Normal has equal intercepts on positive xand yaxes.
Thus slope of the normal is 2sinθ=1
sinθ=12 θ=π6,5π6
But θ5π6 as normal lies in Q1
From (i), Equation of line will be y=x+53
As it touches the ellipse x2a2+y2b2=1,
we have c2=a2m2+b2
Here c=53, m=1
253=a2(1)2+b2
253=a2+b2
3(a2+b2)=25

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