The problem asks us to find something (likely the distance or slope) given two points $P$ and $Q$ in two separate scenarios. For number 17, $P(1, 5)$ and $Q(2, -3)$. For number 18, $P(-1, -5)$ and $Q(2, -3)$. However, the problem does not state what we are trying to calculate. Since the context is missing, I will assume we are asked to find the distance between the two points $P$ and $Q$ for both cases.

1. Problem Description

The problem asks us to find something (likely the distance or slope) given two points $P$ and $Q$ in two separate scenarios.

For number 17, $P(1, 5)$ and $Q(2, -3)$.

For number 18, $P(-1, -5)$ and $Q(2, -3)$.

However, the problem does not state what we are trying to calculate.

Since the context is missing, I will assume we are asked to find the distance between the two points $P$ and $Q$ for both cases.

2. Solution Steps

First, let's derive the distance formula. If we have two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ between them is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

For number 17, $P(1, 5)$ and $Q(2, -3)$.

$x_1 = 1$, $y_1 = 5$, $x_2 = 2$, $y_2 = -3$

$d_{17} = \sqrt{(2 - 1)^2 + (-3 - 5)^2}$

$d_{17} = \sqrt{(1)^2 + (-8)^2}$

$d_{17} = \sqrt{1 + 64}$

$d_{17} = \sqrt{65}$

For number 18, $P(-1, -5)$ and $Q(2, -3)$.

$x_1 = -1$, $y_1 = -5$, $x_2 = 2$, $y_2 = -3$

$d_{18} = \sqrt{(2 - (-1))^2 + (-3 - (-5))^2}$

$d_{18} = \sqrt{(2 + 1)^2 + (-3 + 5)^2}$

$d_{18} = \sqrt{(3)^2 + (2)^2}$

$d_{18} = \sqrt{9 + 4}$

$d_{18} = \sqrt{13}$

3. Final Answer

Assuming the problem asks for the distance between $P$ and $Q$, the answers are:

17) $\sqrt{65}$

18) $\sqrt{13}$