Question

# If 2x+y=5 is tangent at (2,1) on hyperbola intersects asymptotes at A and B such that AB=4√5. If the center of the hyperbola is (−1,−3), then the least possible value of sum of the semi-transverse axis and the semi-conjugate axis is p√q, where HCF(p,q)=1 and q is a prime number, then the value of p+q is

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Solution

## Any tangent to the hyperbola forms a triangle with asymptotes which has constant area ab. Where a & b are the semi-transverse axis and semi-conjugate axis, Area of △PAB=12×h×AB(∵h=length of altitude from P) h=|−2−3−5|√5=10√5=2√5 Ar(△PAB)=12×2√5×4√5=20=ab Using A.M≥G.M a+b2≥√aba+b≥2√20=4√5p+q=9

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