If ab neq 0 and points \((-\mathrm{a}, \mathrm{b})\) and \((-\mathrm{b}, \mathrm{a})\) are in the same quadrant of the xy-plane, is...

Understanding the Question

We need to determine whether point \((-\mathrm{x}, \mathrm{y})\) is in the same quadrant as points \((-\mathrm{a}, \mathrm{b})\) and \((-\mathrm{b}, \mathrm{a})\).

Given Information

• \(\mathrm{ab} \neq 0\) (neither a nor b equals zero)
• Points \((-\mathrm{a}, \mathrm{b})\) and \((-\mathrm{b}, \mathrm{a})\) are in the same quadrant

What We Need to Determine

Two points share a quadrant when:

• Their x-coordinates have the same sign AND
• Their y-coordinates have the same sign

Let's analyze what we know:

• Points \((-\mathrm{a}, \mathrm{b})\) and \((-\mathrm{b}, \mathrm{a})\) are in the same quadrant
• This means: \((-\mathrm{a})\) and \((-\mathrm{b})\) have the same sign, AND b and a have the same sign
• Since \((-\mathrm{a})\) and \((-\mathrm{b})\) have the same sign, then a and b have the same sign

Key Insight

When a and b have the same sign:

• If both are positive → points are in Quadrant II (negative x, positive y)
• If both are negative → points are in Quadrant IV (positive x, negative y)

To determine if \((-\mathrm{x}, \mathrm{y})\) is in the same quadrant, we need to know:

1. The sign of \(-\mathrm{x}\)
2. The sign of \(\mathrm{y}\)
3. Whether these match the quadrant of our reference points

Analyzing Statement 1

Statement 1: \(\mathrm{xy} > 0\)

This means x and y have the same sign (both positive or both negative).

Testing Scenarios

If \(\mathrm{x} > 0\) and \(\mathrm{y} > 0\):

• Then \(-\mathrm{x} < 0\) and \(\mathrm{y} > 0\)
• Point \((-\mathrm{x}, \mathrm{y})\) is in Quadrant II

If \(\mathrm{x} < 0\) and \(\mathrm{y} < 0\):

• Then \(-\mathrm{x} > 0\) and \(\mathrm{y} < 0\)
• Point \((-\mathrm{x}, \mathrm{y})\) is in Quadrant IV

The Problem

We don't know whether a and b are positive or negative, so we can't determine which quadrant contains the reference points. Without knowing if we need \((-\mathrm{x}, \mathrm{y})\) in \(\mathrm{Q2}\) or \(\mathrm{Q4}\), we cannot answer the question.

Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Statement 2: \(\mathrm{ax} > 0\)

This means a and x have the same sign.

What We Can Deduce

• From the question: a and b have the same sign
• From Statement 2: a and x have the same sign
• Therefore: a, b, and x all share the same sign

What's Still Missing

We have no information about y's sign.

Testing possibilities:

• If \(\mathrm{a}, \mathrm{b}, \mathrm{x} > 0\) and \(\mathrm{y} > 0\): \((-\mathrm{x}, \mathrm{y})\) is in \(\mathrm{Q2}\) (same as reference points)
• If \(\mathrm{a}, \mathrm{b}, \mathrm{x} > 0\) and \(\mathrm{y} < 0\): \((-\mathrm{x}, \mathrm{y})\) is in \(\mathrm{Q4}\) (different from reference points)

Since y could be positive or negative, we cannot determine if \((-\mathrm{x}, \mathrm{y})\) is in the same quadrant as the reference points.

Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

Now let's use both statements together.

Combined Information

• From Statement 1: \(\mathrm{xy} > 0\) → x and y have the same sign
• From Statement 2: \(\mathrm{ax} > 0\) → a and x have the same sign
• From the question: a and b have the same sign

Connecting the Pieces

Since:

• a and x have the same sign (Statement 2)
• x and y have the same sign (Statement 1)
• a and b have the same sign (given)

We can conclude: a, b, x, and y all have the same sign

Why This Is Sufficient

Case 1: All four variables are positive

• Reference points \((-\mathrm{a}, \mathrm{b})\) and \((-\mathrm{b}, \mathrm{a})\) → both in \(\mathrm{Q2}\) (negative x, positive y)
• Point \((-\mathrm{x}, \mathrm{y})\) → also in \(\mathrm{Q2}\) (negative x, positive y)
• Answer: YES, they're in the same quadrant

Case 2: All four variables are negative

• Reference points \((-\mathrm{a}, \mathrm{b})\) and \((-\mathrm{b}, \mathrm{a})\) → both in \(\mathrm{Q4}\) (positive x, negative y)
• Point \((-\mathrm{x}, \mathrm{y})\) → also in \(\mathrm{Q4}\) (positive x, negative y)
• Answer: YES, they're in the same quadrant

In both possible cases, \((-\mathrm{x}, \mathrm{y})\) is in the same quadrant as the reference points.

The statements together are SUFFICIENT.

[STOP - Sufficient!] This eliminates choice E.

The Answer: C

Both statements together provide enough information to determine that \((-\mathrm{x}, \mathrm{y})\) is in the same quadrant as the reference points, but neither statement alone is sufficient.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."