Let's go step by step to solve the problem.

(a) Equation of Line PQ

The first step is to find the equation of the line segment $PQ$ joining points $P(2, 3)$ and $Q(0, -1)$.

1. Find the slope of $PQ$: $\text{Slope of } PQ = m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 3}{0 - 2} = \frac{-4}{-2} = 2$

2. Use the point-slope form to find the equation: The equation of a line in point-slope form is given by: $y - y_1 = m(x - x_1)$ Substituting the values of $m = 2$ and point $P(2, 3)$ into the equation: $y - 3 = 2(x - 2)$ Simplifying: $y - 3 = 2x - 4$ $y = 2x - 1$ So, the equation of line $PQ$ is: $y = 2x - 1$

(b) Equation of Line AB

The line $AB$ is perpendicular to $PQ$ and cuts the y-axis at $T(0, 6)$.

1. Find the slope of $AB$: Since $AB$ is perpendicular to $PQ$, the slope of $AB$ will be the negative reciprocal of the slope of $PQ$. $\text{Slope of } AB = -\frac{1}{2}$

2. Use the slope-intercept form to find the equation: The equation of a line in slope-intercept form is given by: $y = mx + c$ Substituting the slope $m = -\frac{1}{2}$ and using the point $T(0, 6)$ (where $c = 6$): $y = -\frac{1}{2}x + 6$ So, the equation of line $AB$ is: $y = -\frac{1}{2}x + 6$

(c) Coordinates of Points R and S

1. Point R (intersection of line PQ with the x-axis): To find the x-coordinate where the line $PQ$ intersects the x-axis, set $y = 0$ in the equation of line $PQ$: $0 = 2x - 1$ Solving for $x$: $x = \frac{1}{2}$ So, the coordinates of $R$ are $\left(\frac{1}{2}, 0\right)$.

2. Point S (intersection of lines PQ and AB): To find the coordinates of $S$, we need to solve the system of equations given by the equations of lines $PQ$ and $AB$: $y = 2x - 1$ $y = -\frac{1}{2}x + 6$ Setting these equal to each other to find the x-coordinate of $S$: $2x - 1 = -\frac{1}{2}x + 6$ Adding $\frac{1}{2}x$ to both sides: $2.5x - 1 = 6$ Adding 1 to both sides: $2.5x = 7$ Solving for $x$: $x = \frac{7}{2.5} = \frac{14}{5} = 2.8$ Substituting $x = 2.8$ back into the equation of $PQ$ to find $y$: $y = 2(2.8) - 1 = 5.6 - 1 = 4.6$ So, the coordinates of $S$ are $(2.8, 4.6)$.

Diagram

Let's create a diagram to visualize the situation.{"size":"1024x1024","prompt":"A geometric diagram showing the line segment PQ joining points P(2, 3) and Q(0, -1) on a Cartesian plane. The line PQ intersects the x-axis at point R (1/2, 0). Another line AB is shown, which is perpendicular to PQ and cuts the y-axis at point T(0, 6). The lines AB and PQ intersect at point S (2.8, 4.6). The axes are labeled, and the key points P, Q, R, T, and S are clearly marked with their respective coordinates."}