Question

If y-intercept of the line $4x - ay = 8$ is thrice its x-intercept, then the value of $a$ is equal to:

• A. $\frac{3}{4}$
• B. $\frac{4}{3}$
• C. $-\frac{3}{4}$
• D. $-\frac{4}{3}$
• E. $-\frac{2}{3}$

Hint:

Always be careful with signs when calculating intercepts. For $Ax + By = C$, if the sign between terms is negative, the intercept associated with that term will carry the sign from the constant $C$ divided by the negative coefficient.

Answer 1

The Correct Option is D

Concept: The intercepts of a line are the points where the line crosses the axes. To find these from a linear equation $Ax + By = C$:
• The x-intercept is found by setting $y = 0$.
• The y-intercept is found by setting $x = 0$.
• Alternatively, convert the equation to the intercept form $\frac{x}{X} + \frac{y}{Y} = 1$, where $X$ is the x-intercept and $Y$ is the y-intercept.

Step 1: Calculate the x-intercept.
Set $y = 0$ in the equation $4x - ay = 8$: \[ 4x - a(0) = 8 \quad \Rightarrow \quad 4x = 8 \quad \Rightarrow \quad x = 2 \] So, the x-intercept is $2$.

Step 2: Calculate the y-intercept.
Set $x = 0$ in the equation $4x - ay = 8$: \[ 4(0) - ay = 8 \quad \Rightarrow \quad -ay = 8 \quad \Rightarrow \quad y = -\frac{8}{a} \] So, the y-intercept is $-\frac{8}{a}$.

Step 3: Use the given condition to solve for $a$.
The problem states that the y-intercept is thrice (3 times) the x-intercept: \[ -\frac{8}{a} = 3 \times 2 \] \[ -\frac{8}{a} = 6 \] Multiplying both sides by $a$ and dividing by $6$: \[ a = -\frac{8}{6} = -\frac{4}{3} \]