Question

The circle passing through $(1,-2)$ and touching the X-axis at$(3,0)$, also passes through the point

• A. $(-5,2)$
• B. $(2,-5)$
• C. $(5,-2)$
• D. $\left(-2,5\right)$

Answer 1

The correct option is C

$(5,-2)$

Explanation for the correct answer:

Finding the equation of the circle:

Given points: $(1,-2)$ and touching the x-axis at$(3,0)$.

The equation of the required circle is ${\left(x-3\right)}^2+y^2+\lambda y=0$

As the point, $(1,-2)$ satisfies the equation, we have

$4+4=2\lambda$ $\Rightarrow \lambda =4$

Therefore the equation of the circle is

$$\begin{gathered} {\left(\mathrm{x}-3\right)}^2+{\mathrm{y}}^2+4\mathrm{y}=0 \\ {\mathrm{x}}^2-6\mathrm{x}+9+{\mathrm{y}}^2+4\mathrm{y}=0 \\ {\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{x}+4\mathrm{y}+9=0 \end{gathered}$$

On trial and error basis, we get $(5,-2)$ satisfies the obtained equation and satisfy the conditions.

Therefore, the correct answer is option (C).