Question

In the \( xy \)-plane, line \( k \) is a line that does not pass through the origin. Which of the following statements individually provide(s) sufficient additional information to determine whether the slope of line \( k \) is negative?

• A. The \( x \)-intercept of line \( k \) is twice the \( y \)-intercept of line \( k \).
• B. The product of the \( x \)-intercept and the \( y \)-intercept of line \( k \) is positive.
• C. Line \( k \) passes through the points \( (a, b) \) and \( (r, s) \), where \( (a-r)(b-s)<0 \).
• D.
• E.

Hint:

For slope questions, remember: intercepts of the same sign imply a negative slope, while intercepts of opposite signs imply a positive slope.

Answer 1

The Correct Option is A

Step 1: Recall slope properties.
For lines not passing through the origin, the slope can be determined from the signs of the \( x \)- and \( y \)-intercepts. If the intercepts have the same sign (both positive or both negative), then the slope of the line is negative. If the intercepts have opposite signs, the slope is positive.
Step 2: Analyze statement (A).
If the \( x \)-intercept is twice the \( y \)-intercept, then both intercepts must have the same sign. Hence the slope of line \( k \) is negative. Statement (A) alone is sufficient.
Step 3: Analyze statement (B).
If the product of the intercepts is positive, then either both intercepts are positive or both are negative. In both cases, the slope is negative. Statement (B) alone is sufficient.
Step 4: Analyze statement (C).
The slope between points \( (a, b) \) and \( (r, s) \) is given by \[ m = \frac{b-s}{a-r}. \] If \( (a-r)(b-s)<0 \), then numerator and denominator have opposite signs, so the slope is negative. Statement (C) alone is sufficient.
Step 5: Conclusion.
Each of the statements (A), (B), and (C) individually provide sufficient information to conclude that the slope of line \( k \) is negative. Hence the correct answer is: \[ \boxed{\text{(A), (B), and (C)}} \]