Find the equation of the circle which circumscribes the triangle formed by the lines $x = 0, y = 0$ and $l x + m y = 1$ .

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Solution

Line: $l x + m y = 1$ cuts $x -$ axis at $A (\frac{1}{l}, 0)$ and $y -$ axis at $B (0, \frac{1}{m})$

Equation of circle circumscribing of triangle $O A B$

$(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$

$\Rightarrow (x - \frac{1}{l}) (x - 0) + (y - 0) (y - \frac{1}{m}) = 0$

$\Rightarrow x^{2} - \frac{x}{l} + y^{2} - \frac{y}{m} = 0$

$∴ x^{2} + y^{2} - \frac{x}{l} - \frac{y}{m} = 0$

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