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Question

If a variable line drawn through the intersection of the lines x3+y4=1 and x4+y3=1, meets the coordinate axes at A and B, (AB), then the locus of the midpoint of AB is:

A
7xy=6(x+y)
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B
14(x+y297(x+y)+168=0
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C
4(x+y)228(x+y)+49=0
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D
6xy=7(x+y)
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Solution

The correct option is A 7xy=6(x+y)
x3+y4=1 and x4+y3=1

These can be simplified as
4x+3y=12 --(1)
3x+4y=12 --(2)
Multiply equation (1) by 3 and (2) by 4,
12x+9y=36
12x+16y=48
Eliminating x, we get
y=127
Putting y in equation (1), x=127
A variable line passes through (127,127) and meets coordinate axis at A and B,

Let the y127=m(x127)

Point A is (127(11m),0)

Point B is (0,127(1m))

midpoint is (67(11m),67(1m))

h=67(11m)

k=67(1m)

Eliminate m, we get locus as
7hk=6(h+k)
7xy=6(x+y)

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