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Question

The lengths of the intercepts made by any circle on the coordinates axes are equal if the centre lies on the line (s) represented by

A
x+y+1=0
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B
x+y=1
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C
x2y2=0
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D
xy=1
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Solution

The correct option is B x2y2=0
Let the equation of any circle be
x2+y2+2gx+2fy+c=0(i)
For intercept made by the circle on x-axis, put y=0 in(i)
x2+2gx+c=0
If x1,x2 are roots of (ii), then length of the intercept on x-axis is
|x1x2|=(x1+x2)24x1x2=2g2c
Similarly length of the intercept of the yaxis is 2f2c
Since the lengths of these intercepts are equal
g2c=f2c g2=f2=(g)2=(f)2
Therefore, centre lies on x2y2=0

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