Question
Question 8, 1.3.39 HW Score SG84\%, 7 of 19 pow Pontssoof 1 Suppose that the point \( (x, y) \) is in the indicated quadrant. Decide whether the given ratio is positive or negative. Recall that \( \mathrm{r}=\sqrt{x^{2}+y^{2}} \). \( 1 \frac{y}{8} \) In Quadrant \( L \) is \( \frac{y}{r} \) positive or negative? Positive Negative
Ask by Todd Gardner. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
In Quadrant I, \( \frac{y}{r} \) is positive.
Solution
In the context of coordinate quadrants, let's clarify the signs of \( y \) and \( r \) in each quadrant to determine whether the ratio \( \frac{y}{r} \) is positive or negative.
### Understanding the Quadrants:
1. **Quadrant I**:
- \( x > 0 \)
- \( y > 0 \)
- \( r = \sqrt{x^2 + y^2} > 0 \)
- **\( \frac{y}{r} \)**: Positive
2. **Quadrant II**:
- \( x < 0 \)
- \( y > 0 \)
- \( r = \sqrt{x^2 + y^2} > 0 \)
- **\( \frac{y}{r} \)**: Positive
3. **Quadrant III**:
- \( x < 0 \)
- \( y < 0 \)
- \( r = \sqrt{x^2 + y^2} > 0 \)
- **\( \frac{y}{r} \)**: Negative
4. **Quadrant IV**:
- \( x > 0 \)
- \( y < 0 \)
- \( r = \sqrt{x^2 + y^2} > 0 \)
- **\( \frac{y}{r} \)**: Negative
### Applying to Your Question:
Assuming **Quadrant I** (since "Quadrant L" seems like a typo and Quadrant I is the first quadrant):
- Both \( y \) and \( r \) are positive.
- Therefore, \( \frac{y}{r} \) is **positive**.
So, **in Quadrant I, \( \frac{y}{r} \) is positive**.
**Answer:** Positive
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Simplify this solution Beyond the Answer
In Quadrant I, both x and y are positive, which means that any ratio that involves y, such as \( \frac{y}{r} \) where r is the distance from the origin, will also be positive. This is because \( r \) is the square root of the sum of the squares of x and y, ensuring it’s always a positive number. With y positive and r positive, the ratio remains positive! Now, it's worth noting that the quadrants are named based on the signs of the coordinates. In Quadrant I, both x and y are greater than zero, leading to positive values throughout. So, any ratios derived from them, like \( \frac{y}{r} \), consistently yield positive outcomes when examined in the context of Quadrant I!
