Question

The relationship between two variables \( x \) and \( y \) is given by \( x + py + q = 0 \) and is shown in the figure. Find the values of \( p \) and \( q \). Note: The figure shown is representative. 

• A. \( p = -\frac{1}{2};\ q = 2 \)
• B. \( p = 2;\ q = -2 \)
• C. \( p = \frac{1}{2};\ q = 4 \)
• D. \( p = 2;\ q = 4 \)

Hint:

When given a line equation with unknowns and a graph, extract two points from the line and substitute into the equation to solve for the constants.

Answer 1

The Correct Option is A

We are given the linear equation:
\[ x + py + q = 0 \quad \Rightarrow \quad y = -\frac{1}{p}x - \frac{q}{p} \]
This is the slope-intercept form: \( y = mx + c \), where:
- Slope \( m = -\frac{1}{p} \)
- Intercept \( c = -\frac{q}{p} \)

From the graph:
- The line passes through the points \( (-2, 0) \) and \( (0, 4) \)

Using these two points, calculate the slope:
\[ m = \frac{4 - 0}{0 - (-2)} = \frac{4}{2} = 2 \quad \Rightarrow \quad -\frac{1}{p} = 2 \quad \Rightarrow \quad p = -\frac{1}{2} \]
Now substitute \( p = -\frac{1}{2} \) into the intercept equation:
\[ y = -\frac{1}{p}x - \frac{q}{p} \quad \Rightarrow \quad y = 2x - 2q \]
We know the line passes through \( (0, 4) \), so:
\[ 4 = 2(0) - 2q \quad \Rightarrow \quad q = -2 \]

Oops! This contradicts the earlier value. Let’s instead directly substitute the known points into the original equation \( x + py + q = 0 \) and solve for \( p \) and \( q \).

From point \( (-2, 0) \):
\[ -2 + p(0) + q = 0 \quad \Rightarrow \quad q = 2 \]
From point \( (0, 4) \):
\[ 0 + p(4) + 2 = 0 \quad \Rightarrow \quad 4p = -2 \quad \Rightarrow \quad p = -\frac{1}{2} \]

Hence, \( p = -\frac{1}{2}, \ q = 2 \) is the correct pair.