The line \( y=a x+b \) passes through \( (1,-2) \) and is perpendicular to the line \( y-3 x=4 \). Find the value of \( a \) and of \( b \).

We are given that the line \[ y = ax + b \] passes through the point \( (1, -2) \) and is perpendicular to the line given by \[ y - 3x = 4. \] **Step 1: Find the slope of the given line.** Starting with \[ y - 3x = 4, \] we solve for \( y \): \[ y = 3x + 4. \] Thus, the slope of this line is \( 3 \). **Step 2: Determine the slope \( a \) of the sought line.** Since the lines are perpendicular, the product of their slopes is \(-1\). Therefore: \[ a \cdot 3 = -1. \] Solving for \( a \): \[ a = -\frac{1}{3}. \] **Step 3: Find the value of \( b \) using the point \( (1, -2) \).** Substitute \( a = -\frac{1}{3} \) and the coordinates of the point into the equation: \[ -2 = \left(-\frac{1}{3}\right)(1) + b. \] This simplifies to: \[ -2 = -\frac{1}{3} + b. \] Isolating \( b \): \[ b = -2 + \frac{1}{3}. \] To combine the terms, write \(-2\) as \(-\frac{6}{3}\): \[ b = -\frac{6}{3} + \frac{1}{3} = -\frac{5}{3}. \] **Final Answer:** \[ a = -\frac{1}{3}, \quad b = -\frac{5}{3}. \]