Circles are drawn passing through the origin O to intersect the coordinate axes at point P and Q such that m. OP + n . OQ is a constant. Show that the circles pass through a fixed point.
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Equation of a circle passing through the origin and having X and Y intercepts equal to a and b respectively is
According to the given condition, we have
ma + nb = k (constant)
i.e. b = 
......
Putting the above value of b in equation , we have, 
i.e. 
which represents the equation of a family of circles passing through the intersection points of the circle
and the line
nx – my = 0.......
Solving equation and , gives the coordinates of the fixed point as 
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