Let equation of line is $\frac{x}{a} + \frac{1}{a} = 1 = \frac{x}{a} + \frac{y}{b} - 1$. If both intercepts are positive, then find the sum of intercepts is equal to?
Step-by-Step Solution
Key Concept: For a line $\frac{x}{a} + \frac{y}{b} = 1$, $a$ and $b$ are the x and y intercepts respectively; analyze the sign conditions to determine validity.
We have $\frac{x}{a} + \frac{y}{b} = 1$ where $a$ and $b$ are intercepts. Case 1: If both intercepts are positive, then $\frac{a}{b} = 4 \Rightarrow ab = 8 - b + a = 4$. This gives $a(4-a) = 8 \Rightarrow a^2 - 4a + 8 = 0$, which has negative discriminant, so no value of $a$ is possible. Case 2: If intercepts are of opposite sign, then $-\frac{ab}{4} = -ab - 8 \Rightarrow b + a = -\frac{4}{4} = -4$. Thus $a(-4-a) = 8 \Rightarrow a^2 + 4a + 8 = 0$, which is possible, hence $b$ is possible. Therefore the sum of intercepts is equal to $-4$.
Correct Answer: 5