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Question

If 2x+y=5 is tangent at (2,1) on hyperbola intersects asymptotes at A and B such that AB=45. 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 pq, where HCF(p,q)=1 and q is a prime number, then the value of p+q is


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=|235|5=105=25
Ar(PAB)=12×25×45=20=ab
Using A.MG.M
a+b2aba+b220=45p+q=9

Mathematics

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