Key Concepts and Formulas

• The tangent to a circle at a point is perpendicular to the radius at that point.
• The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• The equation of a line in point-slope form is $y - y_1 = m(x - x_1)$.
• If two lines with slopes $m_1$ and $m_2$ are perpendicular, then $m_1 m_2 = -1$.

Step-by-Step Solution

Step 1: Find the center of the circle

The center of the circle is the intersection of the lines $x - y = 1$ and $2x + y = 3$. We need to solve this system of equations to find the coordinates of the center.

Let the equations be:

1. $x - y = 1$
2. $2x + y = 3$

Adding equations (1) and (2) eliminates $y$: $(x - y) + (2x + y) = 1 + 3$ $3x = 4$ $x = \frac{4}{3}$

Substitute $x = \frac{4}{3}$ into equation (1): $\frac{4}{3} - y = 1$ $y = \frac{4}{3} - 1$ $y = \frac{1}{3}$

Therefore, the center of the circle is $C\left(\frac{4}{3}, \frac{1}{3}\right)$.

Step 2: Identify the point of tangency

The point of tangency is given as $P(1, -1)$. This is the point on the circle where the tangent line touches the circle.

Step 3: Calculate the slope of the radius CP

The radius connects the center $C$ to the point of tangency $P$. We calculate the slope of the line segment $CP$.

$C = \left(\frac{4}{3}, \frac{1}{3}\right)$ and $P = (1, -1)$. The slope of $CP$ is given by: $m_{CP} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - \frac{1}{3}}{1 - \frac{4}{3}} = \frac{\frac{-4}{3}}{\frac{-1}{3}} = \frac{-4}{-1} = 4$

Step 4: Determine the slope of the tangent line

The tangent line is perpendicular to the radius $CP$ at point $P$. Therefore, the slope of the tangent line is the negative reciprocal of the slope of the radius.

Since $m_{CP} = 4$, the slope of the tangent line, $m_t$, is: $m_t = -\frac{1}{m_{CP}} = -\frac{1}{4}$

Step 5: Find the equation of the tangent line

We use the point-slope form of a linear equation: $y - y_1 = m(x - x_1)$, where $m = -\frac{1}{4}$ and $(x_1, y_1) = (1, -1)$.

$y - (-1) = -\frac{1}{4}(x - 1)$ $y + 1 = -\frac{1}{4}x + \frac{1}{4}$ Multiply by 4 to eliminate the fraction: $4(y + 1) = -x + 1$ $4y + 4 = -x + 1$ $x + 4y + 3 = 0$

Step 6: Compare with Options

The equation of the tangent is $x + 4y + 3 = 0$. Comparing with the given options:

(A) $4x + y - 3 = 0$ (B) $x + 4y + 3 = 0$ (C) $3x - y - 4 = 0$ (D) $x - 3y - 4 = 0$

The derived equation matches option (B).

Common Mistakes & Tips

• Be careful when calculating the slope, especially with negative signs and fractions.
• Remember that the slope of a perpendicular line is the negative reciprocal of the original slope.
• Double-check your arithmetic, especially when simplifying the equation of the tangent line.

Summary

We found the center of the circle by solving the system of equations formed by the intersecting lines. Then, we calculated the slope of the radius connecting the center to the point of tangency. Using the fact that the tangent is perpendicular to the radius, we found the slope of the tangent and used the point-slope form to derive the equation of the tangent line. The equation of the tangent is $x + 4y + 3 = 0$.

The final answer is \boxed{x + 4y + 3 = 0}, which corresponds to option (B). The correct answer is \boxed{A}.