A variable straight line passes through the points of intersection of the lines, x+2y=1 and 2x−y=1 and meets the co-ordinate axes in A and B. The locus of the middle point of AB is
A
x+3y−10xy=0
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B
x−3y−10xy=0
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C
x+3y+10xy=0
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D
None
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Solution
The correct option is Ax+3y−10xy=0
Let the equation of any line passing through the point of intersection of the given line be
(x+2y−1)+a(2x−y−1)=0
Reducing the equation to its intercept form
x(1+2a)(1+a)+y(2−1)(1+a)=1
Therefore coordinates of A and B, where this line meets the coordinate axie respectively.
A=(1+a1+2a,0) on x-axis
B=(0,1+a2−a) on y-axis
Mid point of AB=(1+a2+4a,1+a4−2a)
Now we find the locus of this point by eliminating a between the two expressions