Question

suppose that the point (x, y) is in the indicated quadrant. decide whether the given ratio is positive or negative. recall that ( r = sqrt{x^2 + y^2} ). ii, ( \frac{y}{r} )
in quadrant ii, is ( \frac{y}{r} ) positive or negative?
( \bigcirc ) positive
( \bigcirc ) negative

Explanation:

Step1: Analyze Quadrant II coordinates

In Quadrant II, \( x < 0 \) and \( y > 0 \).

Step2: Analyze \( r \) value

Given \( r=\sqrt{x^{2}+y^{2}} \), since square of real numbers is non - negative and we take the square root, \( r>0 \) (as \( r \) represents the distance from the origin, it can't be zero unless \( x = y=0 \), and in Quadrant II, \( x\) and \( y\) are not both zero).

Step3: Analyze the sign of \( \frac{y}{r} \)

We know that \( y>0 \) (from Quadrant II) and \( r > 0 \). The quotient of a positive number and a positive number is positive. So \( \frac{y}{r}>0 \).

Answer:

Positive