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} ). iii, ( \frac{r}{x} ). choose whether the given ratio is positive or negative. (\bigcirc) positive (\bigcirc) negative

Explanation:

Step1: Analyze r's sign

Since \( r = \sqrt{x^2 + y^2} \), the square root of a sum of squares is always non - negative. And because \( x^2 + y^2>0 \) for a non - zero point \((x,y)\) (and in a quadrant, the point is non - zero), \( r>0 \).

Step2: Analyze x's sign in Quadrant III

In Quadrant III, both the x - coordinate and y - coordinate of a point \((x,y)\) are negative. So \( x<0 \).

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

We know that \( r>0 \) and \( x < 0 \). When we divide a positive number (\(r\)) by a negative number (\(x\)), the result is negative. So \(\frac{r}{x}<0\).

Answer:

Negative