Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) II , y/x

Verified step by step guidance

1. Recall that the ratio given is \( \frac{y}{x} \), where \(x\) and \(y\) are the coordinates of a point in the plane.
2. Identify the signs of \(x\) and \(y\) in Quadrant II. In this quadrant, \(x < 0\) (negative) and \(y > 0\) (positive).
3. Since \(y\) is positive and \(x\) is negative, the ratio \( \frac{y}{x} \) is a positive number divided by a negative number.
4. Dividing a positive number by a negative number results in a negative value, so \( \frac{y}{x} < 0 \) in Quadrant II.
5. Therefore, the ratio \( \frac{y}{x} \) is negative when the point \((x, y)\) lies in Quadrant II.

Key Concepts

Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant II, x is negative and y is positive. Understanding the sign of coordinates in each quadrant helps determine the sign of ratios like y/x.

Sign of Ratios in Different Quadrants

The sign of a ratio such as y/x depends on the signs of y and x individually. Since y is positive and x is negative in Quadrant II, the ratio y/x will be negative. This concept is crucial for evaluating trigonometric ratios based on point location.

Distance from Origin (r = √(x² + y²))

The distance r from the origin to the point (x, y) is always positive and is calculated using the Pythagorean theorem. While r is not directly needed to find the sign of y/x, it is fundamental in defining trigonometric functions and understanding the point's position.