Question

which one of these points lies on the given circle?
(-2.5, -2.5)
(1, 3)
(2, -\sqrt{5})
(-\sqrt{3}, -\sqrt{7})

Explanation:

Step1: Find the circle's equation

The circle has center \((h,k) = (0,0)\) (wait, no, looking at the graph, the center is at \((1.5, 0)\)? Wait, no, the two points on the x - axis are \((0,0)\) and \((3,0)\)? Wait, no, the center is the mid - point of \((0,0)\) and \((3,0)\)? Wait, no, the center is at \((1.5, 0)\)? Wait, no, the radius: the distance from the center (let's find the center first). The circle passes through \((0,0)\) and \((3,0)\), so the center is the mid - point of these two points? Wait, no, the center is at \((1.5, 0)\)? Wait, no, looking at the graph, the center is at \((0,0)\)? No, the blue dot is at \((0,0)\)? Wait, no, the two points: one is \((0,0)\) and the other is \((3,0)\), and the center is at \((1.5, 0)\)? Wait, no, the radius is the distance from the center to \((3,0)\). Wait, maybe I made a mistake. Wait, the standard equation of a circle is \((x - h)^2+(y - k)^2=r^2\). From the graph, the circle is centered at \((0,0)\)? No, the center is at \((1.5, 0)\)? Wait, no, the two points on the x - axis are \((- 3,0)\) and \((3,0)\)? Wait, no, the left intersection with x - axis is at \((- 3,0)\) and right at \((3,0)\)? Wait, no, the graph shows a circle with center at \((0,0)\)? Wait, no, the blue dot is at \((0,0)\)? Wait, the point \((3,0)\) is on the circle. So if the center is \((0,0)\), then the radius \(r = 3\), because the distance from \((0,0)\) to \((3,0)\) is 3. Let's check: the equation of the circle with center \((0,0)\) and radius \(r = 3\) is \(x^{2}+y^{2}=9\).

Step2: Check each point

For point \((-2.5,-2.5)\)

Substitute into \(x^{2}+y^{2}\): \((-2.5)^{2}+(-2.5)^{2}=6.25 + 6.25=12.5
eq9\)

For point \((1,3)\)

Substitute into \(x^{2}+y^{2}\): \(1^{2}+3^{2}=1 + 9 = 10
eq9\)

For point \((2,-\sqrt{5})\)

Substitute into \(x^{2}+y^{2}\): \(2^{2}+(-\sqrt{5})^{2}=4 + 5=9\)

For point \((-\sqrt{3},-\sqrt{7})\)

Substitute into \(x^{2}+y^{2}\): \((-\sqrt{3})^{2}+(-\sqrt{7})^{2}=3 + 7 = 10
eq9\)

Answer:

\((2,-\sqrt{5})\)