find the value of p if the line which passes through (-1, -p) and (-2,2) is parallel to the line 2y+8x-17=0?

Analysis

To solve this problem, we need to use the property of parallel lines. Two lines are parallel if and only if their slopes ($m$) are equal. We will find the slope of the line passing through the two given points and equate it to the slope of the given equation.

Concept: Slope of a Line

1. Slope from two points $(x_1, y_1)$ and $(x_2, y_2)$: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
2. Slope from a linear equation ($Ax + By + C = 0$): Rewrite the equation in the form $y = mx + c$, where $m$ is the slope. Alternatively, $m = -\frac{A}{B}$.

Step-by-Step Solution

Step 1: Find the slope of the given line $2y + 8x - 17 = 0$. Rearrange the equation to make $y$ the subject: $2y = -8x + 17$ Divide by 2: $y = -4x + \frac{17}{2}$ Comparing this to $y = mx + c$, the slope ($m_1$) is -4.

Step 2: Find the slope of the line passing through $(-1, -p)$ and $(-2, 2)$. Using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$: $m_2 = \frac{2 - (-p)}{-2 - (-1)}$ $m_2 = \frac{2 + p}{-2 + 1}$ $m_2 = \frac{2 + p}{-1}$ $m_2 = -(2 + p) = -2 - p$

Step 3: Equate the slopes. Since the lines are parallel, $m_1 = m_2$: $-4 = -2 - p$

Step 4: Solve for $p$. $-4 + 2 = -p$ $-2 = -p$ $p = 2$

Conclusion

The value of $p$ is 2, which corresponds to option D.

Exam Tip

When dealing with parallel lines, always remember: Parallel = Same Slope. If the question had said the lines were perpendicular, you would have used the rule $m_1 \times m_2 = -1$. Also, always be careful with the signs when subtracting negative coordinates—it is the most common place students make mistakes in coordinate geometry!