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let (h,k) be a fixed point where h,k>0.a straight line passing through this point cuts the positive direction of coordinate axes at points P and Q.find the minimum area of the triangle OPQ

5 years ago

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Hi Rahul,


lets take the equation of line to be x/a + y/b = 1 where line intersects x axia at ( a, 0 ) and y axis at (0,b)


Since line passes thru (h,k) therefore h/a + k/b = 1. ------- (1)


A = area of triangle OPQ = 1/2 * a*b


from eqn (1)


A = 1/2 * b * hb / ( b- K ) ( putting value of a from (1) into A )


 = 1/2*b2h/(b - k)


to calculate min value of A, dA/db = 1/2 * h * ( b2 - 2bk ) / ( b-k )2


dA / db = 0 when b2 - 2bk = 0


or b -2k = 0


or b = 2k


At this value of b = 2k , the d2A / db2 is positive that is Minimum of A occurs at this value.


Putting in (1) a = 2h


Thus Minimum value of A occurs when a = 2h and b = 2k


Hence Min(A) = 1/2 * a * b = 1/2 * 2h * 2k = 2hk


Thus Minimum area of traingle OPQ is 2hk.


Please feel free to ask as many questions you have.


Puneet


 


 


 

5 years ago

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