A circle passes through the point (1, -2) and | vedclass

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(C) The equation of a circle touching the $x$-axis at $(h, 0)$ is $(x-h)^2 + (y-k)^2 = k^2$,where $(h, k)$ is the center.Given the circle touches the

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$(5, -2)$

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(C) The equation of a circle touching the $x$-axis at $(h, 0)$ is $(x-h)^2 + (y-k)^2 = k^2$,where $(h, k)$ is the center.Given the circle touches the $x$-axis at $(3, 0)$,the equation is $(x-3)^2 + (y-k)^2 = k^2$.Expanding this,we get $(x-3)^2 + y^2 - 2ky + k^2 = k^2$,which simplifies to $(x-3)^2 + y^2 - 2ky = 0$.Since the circle passes through $(1, -2)$,we substitute these coordinates into the equation:$(1-3)^2 + (-2)^2 - 2k(-2) = 0$$(-2)^2 + 4 + 4k = 0$$4 + 4 + 4k = 0$$8 + 4k = 0$$4k = -8 \Rightarrow k = -2$.Substituting $k = -2$ back into the equation: $(x-3)^2 + y^2 - 2(-2)y = 0 \Rightarrow (x-3)^2 + y^2 + 4y = 0$.Expanding further: $x^2 - 6x + 9 + y^2 + 4y = 0 \Rightarrow x^2 + y^2 - 6x + 4y + 9 = 0$.Checking the options,for $(5, -2)$:$(5)^2 + (-2)^2 - 6(5) + 4(-2) + 9 = 25 + 4 - 30 - 8 + 9 = 38 - 38 = 0$.Thus,the circle passes through $(5, -2)$.

$A$ circle passes through the point $(1, -2)$ and touches the $x$-axis at $(3, 0)$. Through which of the following points does it also pass?

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