Write the equation of the circle having radius 5 and tangent as the line `3x-4y+5=0` at `(1,2)`

Answer

To find the equation of the circle with a radius of 5 that is tangent to the line \(3x - 4y + 5 = 0\) at the point \((1, 2)\), we can follow these steps:

### Step 1: Identify the Tangent Line and the Point of Tangency
The given tangent line is:
\[ 3x - 4y + 5 = 0 \]
and the point of tangency is \((1, 2)\).

### Step 2: Find the Slope of the Tangent Line
To find the slope of the tangent line, we can rewrite the equation in slope-intercept form \(y = mx + b\):
\[ 4y = 3x + 5 \implies y = \frac{3}{4}x + \frac{5}{4} \]
Thus, 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:
\[ m_r = -\frac{1}{m} = -\frac{4}{3} \]

### Step 4: Find the Center of the Circle
Using the point-slope form of the equation of a line, we can find the equation of the radius that passes through the point \((1, 2)\):
\[ y - 2 = -\frac{4}{3}(x - 1) \]
Simplifying this gives:
\[ y - 2 = -\frac{4}{3}x + \frac{4}{3} \implies y = -\frac{4}{3}x + \frac{4}{3} + 2 \]
\[ y = -\frac{4}{3}x + \frac{10}{3} \]

### Step 5: Find the Center Coordinates
Let the center of the circle be \((h, k)\). Since the radius is 5, we can use the distance formula between the center and the point of tangency:
\[ \sqrt{(h - 1)^2 + (k - 2)^2} = 5 \]
Squaring both sides:
\[ (h - 1)^2 + (k - 2)^2 = 25 \]

### Step 6: Substitute the Center in the Line Equation
The center \((h, k)\) must also lie on the line that we derived from the radius. We can substitute \(k\) from the radius equation into the distance equation. From the radius equation:
\[ k = -\frac{4}{3}h + \frac{10}{3} \]
Substituting this into the distance equation:
\[ (h - 1)^2 + \left(-\frac{4}{3}h + \frac{10}{3} - 2\right)^2 = 25 \]
Simplifying \(k - 2\):
\[ k - 2 = -\frac{4}{3}h + \frac{10}{3} - 2 = -\frac{4}{3}h + \frac{10}{3} - \frac{6}{3} = -\frac{4}{3}h + \frac{4}{3} \]
Thus, the distance equation becomes:
\[ (h - 1)^2 + \left(-\frac{4}{3}h + \frac{4}{3}\right)^2 = 25 \]

### Step 7: Solve for \(h\)
Expanding the equation:
\[ (h - 1)^2 + \left(-\frac{4}{3}h + \frac{4}{3}\right)^2 = 25 \]
\[ (h - 1)^2 + \left(\frac{4}{3}(1 - h)\right)^2 = 25 \]
Let’s solve this equation for \(h\).

### Step 8: Find the Circle Equation
Once we have \(h\) and \(k\), we can write the equation of the circle in standard form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \(r = 5\).

### Final Equation of the Circle
After calculating \(h\) and \(k\), we substitute them into the circle equation to get the final answer.