To find the equation of the circles that touch the y-axis at the origin and the line \(5x + 12y - 72 = 0\), we can follow these steps:
### Step 1: Determine the center of the circle
Since the circle touches the y-axis at the origin \((0, 0)\), the center of the circle must be on the x-axis. We can denote the center of the circle as \((h, 0)\).
### Step 2: Find the distance from the center to the line
The distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by the formula:
\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
For our line \(5x + 12y - 72 = 0\), we have \(A = 5\), \(B = 12\), and \(C = -72\). The center of the circle is \((h, 0)\). Thus, the distance from the center to the line is:
\[
d = \frac{|5h + 12(0) - 72|}{\sqrt{5^2 + 12^2}} = \frac{|5h - 72|}{\sqrt{25 + 144}} = \frac{|5h - 72|}{13}
\]
### Step 3: Set the distance equal to the radius
Since the circle touches the line, the radius \(r\) of the circle is equal to the distance from the center to the line. The radius can also be expressed as the distance from the center \((h, 0)\) to the y-axis, which is simply \(h\). Therefore, we have:
\[
h = \frac{|5h - 72|}{13}
\]
### Step 4: Solve the equation
We can solve the equation \(h = \frac{|5h - 72|}{13}\) by considering two cases for the absolute value.
**Case 1:** \(5h - 72 \geq 0\) (i.e., \(h \geq \frac{72}{5}\))
\[
h = \frac{5h - 72}{13}
\]
Multiplying both sides by 13:
\[
13h = 5h - 72
\]
Rearranging gives:
\[
8h = -72 \implies h = -9
\]
Since \(h = -9\) does not satisfy \(h \geq \frac{72}{5}\), we discard this solution.
**Case 2:** \(5h - 72 < 0\) (i.e., \(h < \frac{72}{5}\))
\[
h = \frac{-(5h - 72)}{13} = \frac{-5h + 72}{13}
\]
Multiplying both sides by 13:
\[
13h = -5h + 72
\]
Rearranging gives:
\[
18h = 72 \implies h = 4
\]
### Step 5: Find the radius
Now that we have \(h = 4\), we can find the radius:
\[
r = h = 4
\]
### Step 6: Write the equation of the circle
The equation of the circle with center \((h, 0) = (4, 0)\) and radius \(r = 4\) is:
\[
(x - 4)^2 + (y - 0)^2 = 4^2
\]
This simplifies to:
\[
(x - 4)^2 + y^2 = 16
\]
### Step 7: Check for other possible centers
We also need to check for the other possible center by considering the negative case of the absolute value:
\[
h = \frac{72 - 5h}{13}
\]
Multiplying both sides by 13:
\[
13h = 72 - 5h
\]
Rearranging gives:
\[
18h = 72 \implies h = 4
\]
This confirms our previous result.
### Final Answer
Thus, the equations of the circles that touch the y-axis at the origin and the line \(5x + 12y - 72 = 0\) are:
1. \((x - 4)^2 + y^2 = 16\)
2. \((x + 4)^2 + y^2 = 16\)
