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(A) The equation of a circle touching both axes in the first quadrant is $(x - a)^2 + (y - a)^2 = a^2$,which simplifies to $x^2 + y^2 - 2ax - 2ay + a^

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$x^2 + y^2 - 2x - 2y + 1 = 0, \; x^2 + y^2 - 10x - 10y + 25 = 0$

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(A) The equation of a circle touching both axes in the first quadrant is $(x - a)^2 + (y - a)^2 = a^2$,which simplifies to $x^2 + y^2 - 2ax - 2ay + a^2 = 0$.Since the circle passes through the point $(1, 2)$,we substitute these coordinates into the equation:$(1 - a)^2 + (2 - a)^2 = a^2$$1 - 2a + a^2 + 4 - 4a + a^2 = a^2$$a^2 - 6a + 5 = 0$$(a - 1)(a - 5) = 0$Thus,$a = 1$ or $a = 5$.For $a = 1$,the equation is $x^2 + y^2 - 2x - 2y + 1 = 0$.For $a = 5$,the equation is $x^2 + y^2 - 10x - 10y + 25 = 0$.

The equations of the circles touching both the axes and passing through the point $(1, 2)$ are

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