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Question

The line 2x+y=1 is tangent to the hyperbola x2a2y2b2=1. If this line pasess through the point of intersection of the nearest directrix and the X-axis, then the eccentricity of the hyperbola is ___


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Solution

On substituting (ae,0) in y=2x+1, we get
0=2ae+1
ae=12
Also, y=2x+1 is tangent to hyperbola.
1=4a2b21a2=4(e21)4e2=5e2
e45e2+4=0(e24)(e21)=0
e=2,e=1

e = 1 gives the conic as parabola. But conic is given as hyperbola, hence e = 2.


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