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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 x2−y2=0 Let the equation of any circle bex2+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|x1−x2|=√(x1+x2)2−4x1x2=2√g2−c Similarly length of the intercept of the yaxis is 2√f2−c Since the lengths of these intercepts are equal√g2−c=√f2−c ⇒g2=f2=(−g)2=(−f)2Therefore, centre lies on x2−y2=0

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