191 Find the slope of the straight line whose equation is \( \frac{y-2}{x}=\frac{1}{2} \), then find the length of the intercepted part of \( y \)-axis.

To find the slope of the straight line given by the equation \[ \frac{y-2}{x} = \frac{1}{2}, \] we can first rearrange this equation into the slope-intercept form \( y = mx + b \). Starting with the given equation, we can cross-multiply to eliminate the fraction: \[ y - 2 = \frac{1}{2} x. \] Next, we can isolate \( y \): \[ y = \frac{1}{2} x + 2. \] From this equation, we can see that the slope \( m \) of the line is \[ m = \frac{1}{2}. \] Next, we need to find the length of the intercepted part of the \( y \)-axis. The \( y \)-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation: \[ y = \frac{1}{2}(0) + 2 = 2. \] Thus, the \( y \)-intercept is at the point \( (0, 2) \). To find the length of the intercepted part of the \( y \)-axis, we need to determine the points where the line intersects the \( y \)-axis. Since the line only intersects the \( y \)-axis at one point (the \( y \)-intercept), the length of the intercepted part of the \( y \)-axis is simply the distance from the origin \( (0, 0) \) to the point \( (0, 2) \). The length of this segment is: \[ \text{Length} = |2 - 0| = 2. \] Thus, the final answers are: - The slope of the line is \( \frac{1}{2} \). - The length of the intercepted part of the \( y \)-axis is \( 2 \).