25. Find the equation of the line tangent to circle \( S \) centered at \( (1,1) \) if the point of tangency is \( (5,4) \).

To find the equation of the tangent line to the circle \( S \) centered at \( (1,1) \) at the point of tangency \( (5,4) \), follow these steps: 1. **Determine the Slope of the Radius:** The radius of the circle \( S \) goes from the center \( (1,1) \) to the point of tangency \( (5,4) \). \[ \text{Slope of the radius} = \frac{4 - 1}{5 - 1} = \frac{3}{4} \] 2. **Find the Slope of the Tangent Line:** The tangent line is perpendicular to the radius at the point of tangency. Therefore, its slope is the negative reciprocal of the slope of the radius. \[ \text{Slope of the tangent} = -\frac{4}{3} \] 3. **Use the Point-Slope Form to Find the Equation:** With the slope \( -\frac{4}{3} \) and the point \( (5,4) \), the equation of the tangent line in point-slope form is: \[ y - 4 = -\frac{4}{3}(x - 5) \] Simplifying this, we get: \[ y = -\frac{4}{3}x + \frac{32}{3} \] To express it in standard form: \[ 4x + 3y = 32 \] **Final Answer:** After simplifying, an equation for the tangent line is 4 x + 3 y = 32.