Find the equation of circle which circumscribe the triangle formed by the line x = 0 , y = 0 and Ix + my = 1 Find the locus of its center if $l^{2} + m^{2} = 4 l^{2} m^{2}$

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Solution

Circle OAB in usual is
$x^{2} + y^{2} - (\frac{1}{l}) x - (\frac{1}{m}) y = 0$ whose centre is
(x , y) = $(\frac{1}{2 l}, \frac{1}{2 m})$ .The given condition can be written as
$\frac{1}{l^{2}} + \frac{1}{m^{2}} = 4$
$∴$ Locus is $(2 x)^{2} + (2 y)^{2}$ = 4 or $x^{2} + y^{2}$ = 1

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