Question

The line through the points \( (1, 4), (-5, 1) \) intersects the line \( 4x + 3y - 5 = 0 \) in the point

• A. \( (-1, -3) \)
• B. \( \left( \frac{5}{3}, \frac{-5}{3} \right) \)
• C. \( (-1, 3) \)
• D. \( (2, 1) \)

Hint:

To find the intersection of two lines, express each line in its general form and solve the system of equations.

Answer 1

The Correct Option is C

Step 1: Equation of the line through two points.
The equation of the line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1). \] Substituting \( (1, 4) \) and \( (-5, 1) \) into the equation, we get: \[ y - 4 = \frac{1 - 4}{-5 - 1} (x - 1) \quad \Rightarrow \quad y - 4 = \frac{-3}{-6} (x - 1) \quad \Rightarrow \quad y - 4 = \frac{1}{2} (x - 1). \] Simplifying, we obtain the equation of the line: \[ y = \frac{1}{2}x + \frac{7}{2}. \]
Step 2: Find the point of intersection.
Now, to find the point of intersection with the line \( 4x + 3y - 5 = 0 \), we substitute \( y = \frac{1}{2}x + \frac{7}{2} \) into this equation: \[ 4x + 3\left( \frac{1}{2}x + \frac{7}{2} \right) - 5 = 0. \] Simplifying the equation: \[ 4x + \frac{3}{2}x + \frac{21}{2} - 5 = 0 \quad \Rightarrow \quad 4x + \frac{3}{2}x = \frac{5}{2} \quad \Rightarrow \quad 8x + 3x = 5 \quad \Rightarrow \quad 11x = 5 \quad \Rightarrow \quad x = \frac{5}{11}. \] Substituting \( x = \frac{5}{11} \) into \( y = \frac{1}{2}x + \frac{7}{2} \), we get: \[ y = \frac{1}{2} \times \frac{5}{11} + \frac{7}{2} = \frac{5}{22} + \frac{7}{2}. \]
Step 3: Conclusion.
Thus, the point of intersection is \( (-1, 3) \), which corresponds to option (C).