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{y}{r}. choose whether the given ratio is positive or negative. negative positive

Explanation:

Step1: Recall quadrant III properties

In quadrant III, \(x<0\) and \(y < 0\).

Step2: Recall the formula for \(r\)

We know that \(r=\sqrt{x^{2}+y^{2}}\), and since \(x^{2}\geq0\) and \(y^{2}\geq0\), \(r> 0\) for any non - zero point \((x,y)\) (and \(r = 0\) only when \(x = 0\) and \(y=0\)).

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

We have \(y<0\) and \(r>0\). A negative number divided by a positive number is negative. So \(\frac{y}{r}<0\).

Answer:

Negative