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} )

1. ( \frac{r}{y} )

in quadrant ii, is ( \frac{r}{y} ) positive or negative?
( \bigcirc ) negative
( \bigcirc ) positive

Explanation:

Step1: Analyze r's sign

Since \( r=\sqrt{x^{2}+y^{2}} \), and the square root of a sum of squares is always non - negative. Also, \( x^{2}+y^{2}>0 \) for a non - zero point \((x,y)\), so \( r>0 \) (because if \( x = 0 \) and \( y=0 \), it's not a point in a quadrant, so for a point in Quadrant II, \( x
eq0 \) and \( y
eq0 \), so \( r=\sqrt{x^{2}+y^{2}}>0\)).

Step2: Analyze y's sign in Quadrant II

In Quadrant II, the x - coordinate is negative and the y - coordinate is positive. So for a point \((x,y)\) in Quadrant II, \( y>0 \).

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

We know that \( r>0 \) and \( y > 0 \). When we divide two positive numbers, the result is positive. So \(\frac{r}{y}=\frac{\text{positive}}{\text{positive}}=\text{positive}\).

Answer:

Positive