First slide

Definition of a circle

Question

The straight line $\frac{x}{a} + \frac{y}{b} = 1$ cuts the coordinate axes at
$A$ and $B .$ The equation of the circle passing through O ( 0, 0),
$A$ and $B,$ is

Moderate

A

$x^{2} + y^{2} - a x - b y = 0$
B

$x^{2} + y^{2} - 2 a x - 2 b y = 0$
C

$x^{2} + y^{2} + a x + b y = 0$
D

$x^{2} + y^{2} = a^{2} + b^{2}$

Solution

The straight line $\frac{x}{a} + \frac{y}{b} = 1$ cuts the coordinate axes at $A$ $(a, 0)$ and $B$ $(0, b) .$
Let $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ (i)
be the circle passing through $O,$ $A$ and $B .$ Then,
$\begin{array}{l} 0 + c = 0 (ii) \\ a^{2} + 2 g a + c = 0 (iii) \\ b^{2} + 2 f b + c = 0 (iv) \end{array}$
Solving (ii), (iii) and (iv), we obtain
$g = - \frac{a}{2}, f = - \frac{b}{2}$ and $c = 0$
Substituting these values in (i), we obtain the equation of the
required circle as $x^{2} + y^{2} - a x - b y = 0$

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