Question

point a lies on the circle and has an x - coordinate of 1. which is the correct calculation of the y - coordinate of point a? \\(\sqrt{(0 - 1)^2+(0 - y)^2}=2\\) \\(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\\) \\(\sqrt{(0 - 0)^2+(1 - y)^2}=2^2\\)

Explanation:

Step1: Identify the circle's center and radius

The circle is centered at \((0,0)\) (from the graph) and passes through \((2,0)\), so the radius \(r = 2\). The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For point \(A(1,y)\) on the circle, the distance from the center \((0,0)\) to \(A\) should equal the radius.

Step2: Apply the distance formula

Using the distance formula with \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(1,y)\), we get \(\sqrt{(0 - 1)^2+(0 - y)^2}\). This distance must equal the radius, which is \(2\). But wait, the distance formula squared is also used in the circle equation \((x - h)^2+(y - k)^2=r^2\) (where \((h,k)\) is the center). So the distance from \((0,0)\) to \((1,y)\) is \(\sqrt{(0 - 1)^2+(0 - y)^2}\), and since the radius is \(2\), squaring both sides (from the circle equation \(x^2 + y^2=r^2\)) gives \((0 - 1)^2+(0 - y)^2 = 2^2\), which is equivalent to \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\) (taking square roots, but actually the correct equation from the circle formula is \((x - 0)^2+(y - 0)^2 = 2^2\), so for \(x = 1\), \(1^2+y^2 = 4\), and solving for \(y\) would involve \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\) (since the left side is the distance, equal to radius \(2\)). Wait, no—wait, the second option has \(2^2\) on the right. Wait, let's re - examine. The circle equation is \((x - h)^2+(y - k)^2=r^2\). Here, \(h = 0\), \(k = 0\), \(r = 2\), so \((x)^2+(y)^2=4\). For point \(A(1,y)\), substituting \(x = 1\) gives \(1 + y^2=4\), which can be written as \((0 - 1)^2+(0 - y)^2=2^2\). If we take the square root of both sides, \(\sqrt{(0 - 1)^2+(0 - y)^2}=\sqrt{2^2}=2\). Wait, but the options: the first option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\), the second is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\), the third is \(\sqrt{(0 - 0)^2+(1 - y)^2}=2^2\).

Wait, the distance from \((0,0)\) to \((1,y)\) is \(\sqrt{(1 - 0)^2+(y - 0)^2}=\sqrt{(0 - 1)^2+(0 - y)^2}\), and this distance is the radius, which is \(2\). But in the circle equation, \((x - 0)^2+(y - 0)^2=r^2\), so \((1)^2+(y)^2 = 2^2\), which is \((0 - 1)^2+(0 - y)^2=2^2\). If we take the square root of both sides, \(\sqrt{(0 - 1)^2+(0 - y)^2}=\sqrt{2^2}=2\). But the second option has \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\), which is wrong. Wait, no—maybe I made a mistake. Wait, the first option: \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\). Let's calculate the left - hand side: \(\sqrt{(- 1)^2+(-y)^2}=\sqrt{1 + y^2}\). Setting this equal to \(2\) gives \(1 + y^2=4\), so \(y^2 = 3\), \(y=\pm\sqrt{3}\). The second option: \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2 = 4\). Then squaring both sides, \(1 + y^2=16\), which is not correct. The third option: \(\sqrt{(0 - 0)^2+(1 - y)^2}=2^2\), which is \(\sqrt{0+(1 - y)^2}=4\), so \((1 - y)^2 = 16\), which is also incorrect. Wait, so the first option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\), which is correct because the distance from \((0,0)\) to \((1,y)\) is the radius \(2\). But wait, the problem's options: maybe I misread. Wait, the first option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\), the second is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\), the third is \(\sqrt{(0 - 0)^2+(1 - y)^2}=2^2\). So the correct one is the second option? Wait, no—wait, the circle equation is \((x)^2+(y)^2=r^2\), so \(x = 1\), \(y = y\), \(r = 2\), so \(1 + y^2=4\), which is \((0 - 1)^2+(0 - y)^2=2^2\). If we take the square root of both sides, \(\sqrt{(0 - 1)^2+(0 - y)^2}=\sqrt{2^2}=2\). But the second option has \(2^2\) on the right. Wait, maybe the question is about the equation to solve for \(y\). Let's think again. The distance between \((0,0)\) and \((1,y)\) is equal to the radius, which is \(2\). The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), so \(d=\sqrt{(1 - 0)^2+(y - 0)^2}=\sqrt{(0 - 1)^2+(0 - y)^2}\). This distance \(d\) is equal to the radius \(2\), so \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\). But the second option has \(2^2\) on the right. Wait, no—maybe the options are written with the circle equation in mind. The circle equation is \((x - 0)^2+(y - 0)^2=2^2\), so for \(x = 1\), we have \((1)^2+(y)^2=2^2\), which can be rewritten as \(\sqrt{(1 - 0)^2+(y - 0)^2}=2\) (since the left side is the square root of the left side of the circle equation, and the right side is the square root of \(2^2\), which is \(2\)). But the second option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\), which would mean \(\sqrt{(0 - 1)^2+(0 - y)^2}=4\), which is wrong. Wait, I must have made a mistake. Wait, looking at the graph, the circle goes from \((- 2,0)\) to \((2,0)\)? No, the graph shows the circle passing through \((2,0)\) and \((0,2)\) etc. Wait, the center is \((0,0)\), and the point \((2,0)\) is on the circle, so radius is \(2\). So the circle equation is \(x^{2}+y^{2}=4\). For point \(A\) with \(x = 1\), substitute into the equation: \(1^{2}+y^{2}=4\), so \(y^{2}=4 - 1=3\), so \(y=\pm\sqrt{3}\). To get this, we start with the distance from \((0,0)\) to \((1,y)\) is \(2\), so \(\sqrt{(1 - 0)^{2}+(y - 0)^{2}}=2\), which is \(\sqrt{(0 - 1)^{2}+(0 - y)^{2}}=2\) (since \((1 - 0)^2=(0 - 1)^2\) and \((y - 0)^2=(0 - y)^2\)). So the first option is correct? But wait, the second option has \(2^2\) on the right. Wait, no—maybe the question is using the circle equation in the form of the distance formula squared. The distance formula squared is \((x_2 - x_1)^2+(y_2 - y_1)^2=d^2\). So if the distance \(d = 2\), then \(d^2=4\), so \((0 - 1)^2+(0 - y)^2=2^2\), which is the equation, and if we take the square root of both sides, we get \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\). But the second option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\), which is \(\sqrt{(0 - 1)^2+(0 - y)^2}=4\), which is wrong. Wait, the first option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\), which is correct because the distance from \((0,0)\) to \((1,y)\) is \(2\) (the radius). The second option is incorrect, the third is incorrect. Wait, but maybe I misread the options. Let me check again:

Option 1: \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\)

Option 2: \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\)

Option 3: \(\sqrt{(0 - 0)^2+(1 - y)^2}=2^2\)

So the correct equation to find \(y\) is derived from the distance formula (distance from center \((0,0)\) to \(A(1,y)\) is radius \(2\)), so \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\), which is option 1? But wait, the circle equation is \(x^{2}+y^{2}=r^{2}\), so \(x = 1\), \(y = y\), \(r = 2\), so \(1 + y^{2}=4\), and taking square roots of both sides (after moving \(1\) to the other side) would be \(y=\pm\sqrt{4 - 1}\), but the equation to solve for \(y\) starts with \(\sqrt{(1)^{2}+(y)^{2}}=2\), which is the same as \(\sqrt{(0 - 1)^{2}+(0 - y)^{2}}=2\). So the first option is correct. Wait, but maybe the options are written with a typo, and the second option is supposed to be \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\) but with \(2^2\) as a mistake. No, let's re - evaluate. The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and \(d = r = 2\), so \(d^2=r^2\), so \((x_2 - x_1)^2+(y_2 - y_1)^2=r^2\). So substituting \((x_1,y_1)=(0,0)\), \((x_2,y_2)=(1,y)\), and \(r = 2\), we get \((1 - 0)^2+(y - 0)^2=2^2\), which is \((0 - 1)^2+(0 - y)^2=2^2\). If we take the square root of both sides, we get \(\sqrt{(0 - 1)^2+(0 - y)^2}=\sqrt{2^2}=2\). But the second option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\), which is \(\sqrt{(0 - 1)^2+(0 - y)^2}=4\), which is wrong. The first option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\), which is correct. Wait, but maybe the question is considering the circle equation in the form of the distance formula without squaring. I think the correct option is the second one? No, I'm confused. Wait, let's do the math. If we use the first option: \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\). Squaring both sides: \((0 - 1)^2+(0 - y)^2=4\), so \(1 + y^{2}=4\), so \(y^{2}=3\), \(y=\pm\sqrt{3}\), which is correct. If we use the second option: \(\sqrt{(0 - 1)^2+(0 - y)^2}=4\). Squaring both sides: \(1 + y^{2}=16\), \(y^{2}=15\), \(y=\pm\sqrt{15}\), which is wrong. The third option: \(\sqrt{(0 - 0)^2+(1 - y)^2}=4\), squaring: \(0+(1 - y)^2=16\), \((1 - y)^2=16\), \(1 - y=\pm4\), \(y = 1\pm4\), which is wrong. So the correct option is the first one? Wait, but the problem's second option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\). Wait, maybe I misread the options. Oh! Wait, maybe the first option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\) and the second is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\) (i.e., \(4\)). But according to the circle equation, the correct equation to solve for \(y\) is \((1)^2+y^2=2^2\), which is equivalent to \(\sqrt{(1)^2+y^2}=2\), so the first option is correct.

Answer:

The correct calculation is \(\boldsymbol{\sqrt{(0 - 1)^2+(0 - y)^2}=2}\) (the first option among the given options, assuming the first option is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2\), the second is \(\sqrt{(0 - 1)^2+(0 - y)^2}=2^2\), and the third is \(\sqrt{(0 - 0)^2+(1 - y)^2}=2^2\)).