A variable line drawn through the point of intersection of the lines xa+yb=1,xb+ya=1 meets the coordinate axes in A and B. Then the locus of the mid point of AB is
A
2xy(a+b)=ab(x+y)
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B
xy(a+b)=ab(x−y)
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C
xy(a+b)=ab(x+y)
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D
xy(a+b)=2ab(x+y)
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Solution
The correct option is A2xy(a+b)=ab(x+y) Let C be the point of intersection of the lines.
Then, C=(aba+b+aba+b) Let M(h,k) be the mid- point of AB Then, the equation of A may be written as x2h+y2k=1 Since AB passes through C, we have ab2h(a+b)+ab2k(a+b)=1⇒1h+1k=2(a+b)ab Therefore locus of M(h,k) is 1x+1y=2(a+b)ab