Find the equation of the circle which has radius 5 units and which is tangent to the line `3x+4y-16=0` at the point (4, 1).

Answer

To find the equation of the circle with a radius of 5 units that is tangent to the line \(3x + 4y - 16 = 0\) at the point (4, 1), we can follow these steps:

### Step 1: Identify the point of tangency and the radius
The point of tangency is given as (4, 1). The radius of the circle is 5 units.

### Step 2: Find the slope of the tangent line
The equation of the tangent line is \(3x + 4y - 16 = 0\). We can rewrite this in slope-intercept form \(y = mx + b\):
\[ 4y = -3x + 16 \implies y = -\frac{3}{4}x + 4 \]
The slope \(m\) of the tangent line is \(-\frac{3}{4}\).

### Step 3: Determine the slope of the radius
The radius of the circle at the point of tangency is perpendicular to the tangent line. Therefore, the slope of the radius \(m_r\) is the negative reciprocal of the slope of the tangent line:
\[ m_r = \frac{4}{3} \]

### Step 4: Find the center of the circle
Let the center of the circle be \((h, k)\). Since the radius is 5 units, we can use the point-slope form of the line to find the center:
\[ k - 1 = \frac{4}{3}(h - 4) \]
This simplifies to:
\[ k - 1 = \frac{4}{3}h - \frac{16}{3} \]
Multiplying through by 3 to eliminate the fraction:
\[ 3k - 3 = 4h - 16 \implies 4h - 3k = 13 \quad \text{(Equation 1)} \]

### Step 5: Use the distance formula
The distance from the center \((h, k)\) to the point of tangency \((4, 1)\) must equal the radius:
\[ \sqrt{(h - 4)^2 + (k - 1)^2} = 5 \]
Squaring both sides:
\[ (h - 4)^2 + (k - 1)^2 = 25 \]
Expanding this:
\[ (h^2 - 8h + 16) + (k^2 - 2k + 1) = 25 \]
This simplifies to:
\[ h^2 + k^2 - 8h - 2k + 17 = 25 \implies h^2 + k^2 - 8h - 2k - 8 = 0 \quad \text{(Equation 2)} \]

### Step 6: Solve the equations
Now we have two equations:
1. \(4h - 3k = 13\) (Equation 1)
2. \(h^2 + k^2 - 8h - 2k - 8 = 0\) (Equation 2)

From Equation 1, we can express \(k\) in terms of \(h\):
\[ k = \frac{4h - 13}{3} \]
Substituting this into Equation 2:
\[ h^2 + \left(\frac{4h - 13}{3}\right)^2 - 8h - 2\left(\frac{4h - 13}{3}\right) - 8 = 0 \]
Expanding and simplifying this equation will yield values for \(h\) and subsequently for \(k\).

### Step 7: Find the equation of the circle
Once we find the center \((h, k)\), we can use the standard form of the circle's equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \(r = 5\).