Equations of circles which pass through the points (1,-2) and (3,-4) and touch the x-axis is

Options

A. `x^(2)+y^(2)+6x+2y+9=0`
B. `x^(2)+y^(2)+10x+20y+25=0`
C. `x^(2)+y^(2)-6x+4y+9=0`
D. none

Answer

To find the equations of circles that pass through the points (1, -2) and (3, -4) and touch the x-axis, we can follow these steps:

### Step 1: General Equation of the Circle
The general equation of a circle that touches the x-axis can be expressed as:
\[ (x - h)^2 + (y - k)^2 = k^2 \]
where \((h, k)\) is the center of the circle, and \(k\) is the radius (since the circle touches the x-axis, the radius is equal to the y-coordinate of the center).