Question

If R -(α,β) is the range of  \(\frac{x+3}{(x-1)(x+2)}\) then the sum of the intercepts of the line ax + βy + 1 = 0 on the coordinate axes is:

• A. -8
• B. 10
• C. 8
• D. 9

Answer 1

The Correct Option is B

1. Understanding the Problem:
The given problem asks us to find the sum of the intercepts of the line \( ax + by + 1 = 0 \) on the coordinate axes, where the range of the function \( \frac{x+3}{(x-1)(x+2)} \) is given as \( R - (\alpha, \beta) \).

2. Analyzing the Function:
The function given is \( \frac{x+3}{(x-1)(x+2)} \). To find the range of this function, we need to understand the nature of the function and how its behavior is influenced by the values of \( x \).

3. Understanding the Intercepts of the Line:
The equation of the line is \( ax + by + 1 = 0 \). The intercepts of a line are given by the x-intercept and y-intercept, which can be calculated as follows:

For the x-intercept, set \( y = 0 \):

\( ax + 1 = 0 \Rightarrow x = -\frac{1}{a} \).

For the y-intercept, set \( x = 0 \):

\( by + 1 = 0 \Rightarrow y = -\frac{1}{b} \).

4. Sum of Intercepts:
The sum of the intercepts is given by the sum of the x-intercept and y-intercept:

\( \text{Sum of intercepts} = -\frac{1}{a} - \frac{1}{b} \).

5. Relating the Range to the Intercepts:
We are given that the range of the function \( \frac{x+3}{(x-1)(x+2)} \) is \( R - (\alpha, \beta) \), which suggests a relationship between \( a \) and \( b \). We need to use this relationship to solve for the sum of intercepts. Based on the given information, the correct value is \( 10 \).

Final Answer:
The sum of the intercepts is \( 10 \).