4.3 Q-4

Question Statement

Find the value of $K$ such that:

1. The line joining points $A(7,3)$ and $B(K,-6)$ is parallel to the line joining points $C(-4,5)$ and $D(-6,4)$.
2. The line joining points $A(7,3)$ and $B(K,-6)$ is perpendicular to the line joining points $C(-4,5)$ and $D(-6,4)$.

---

Background and Explanation

To solve this, we need to use the concept of the slope of a line. The slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• Parallel Lines: Two lines are parallel if their slopes are equal, i.e., $m_1 = m_2$.
• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is $-1$, i.e., $m_1 \cdot m_2 = -1$.

---

Solution

Step 1: Calculate the slope of line $\overline{CD}$

The points $C(-4,5)$ and $D(-6,4)$ are given. Using the slope formula:

$$m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 5}{-6 - (-4)} = \frac{-1}{-2} = \frac{1}{2}$$

So, the slope of $\overline{CD}$ is $m_2 = \frac{1}{2}$.

Step 2: Calculate the slope of line $\overline{AB}$

The points $A(7,3)$ and $B(K,-6)$ are given. The slope of $\overline{AB}$ is:

$$m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 3}{K - 7} = \frac{-9}{K - 7}$$

We now have an expression for $m_1$, the slope of line $\overline{AB}$, in terms of $K$.

Part (i): Find $K$ for Parallel Lines

For the lines $\overline{AB}$ and $\overline{CD}$ to be parallel, their slopes must be equal. Therefore, we set $m_1 = m_2$:

$$\frac{-9}{K - 7} = \frac{1}{2}$$

Now, solve for $K$:

$$\begin{aligned} \frac{-9}{K - 7} &= \frac{1}{2} -9 \cdot 2 &= (K - 7) \cdot 1 -18 &= K - 7 K &= -18 + 7 K &= -11 \end{aligned}$$

Thus, the value of $K$ for the lines to be parallel is $K = -11$.

Part (ii): Find $K$ for Perpendicular Lines

For the lines $\overline{AB}$ and $\overline{CD}$ to be perpendicular, the product of their slopes must be $-1$. This gives the equation:

$$m_1 \cdot m_2 = -1$$

Substitute $m_1 = \frac{-9}{K - 7}$ and $m_2 = \frac{1}{2}$:

$$\frac{-9}{K - 7} \cdot \frac{1}{2} = -1$$

Now, solve for $K$:

$$\begin{aligned} \frac{-9}{2(K - 7)} &= -1 -9 &= -2(K - 7) 9 &= 2(K - 7) 9 &= 2K - 14 2K &= 9 + 14 2K &= 23 K &= \frac{23}{2} \end{aligned}$$

Thus, the value of $K$ for the lines to be perpendicular is $K = \frac{23}{2}$.

---

Key Formulas or Methods Used

• Slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• For parallel lines, $m_1 = m_2$.
• For perpendicular lines, $m_1 \cdot m_2 = -1$.

---

Summary of Steps

1. Calculate the slope of line $\overline{CD}$ using the formula: $m_2 = \frac{1}{2}$.
2. Set up the equation for the slope of line $\overline{AB}$: $m_1 = \frac{-9}{K - 7}$.
3. For parallel lines: Set $m_1 = m_2$ and solve for $K$. The result is $K = -11$.
4. For perpendicular lines: Set $m_1 \cdot m_2 = -1$ and solve for $K$. The result is $K = \frac{23}{2}$.