Question

does the point $(1,\sqrt{7})$ lie on the circle shown? explain. yes, the distance from $(-2,4)$ to $(1,\sqrt{7})$ is 4 units. yes, the distance from $(-2,0)$ to $(1,\sqrt{7})$ is 4 units. no, the distance from $(-2,0)$ to $(1,\sqrt{7})$ is not 4 units. no, the distance from $(-2,4)$ to $(1,\sqrt{7})$ is not 4 units. distance formula: $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

Explanation:

Step1: Find the center and radius of the circle

From the graph, the center of the circle is \((- 2,0)\). The radius \(r\) is the distance from \((-2,0)\) to \((-2,4)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1=-2,y_1 = 0,x_2=-2,y_2 = 4\), then \(d=\sqrt{(-2+2)^2+(4 - 0)^2}=\sqrt{0 + 16}=4\). So the radius \(r = 4\).

Step2: Calculate the distance from the center \((-2,0)\) to the point \((1,\sqrt{7})\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(x_1=-2,y_1 = 0,x_2=1,y_2=\sqrt{7}\). Then \(d=\sqrt{(1+2)^2+(\sqrt{7}-0)^2}=\sqrt{9 + 7}=\sqrt{16}=4\)

Answer:

Yes, the distance from \((-2,0)\) to \((1,\sqrt{7})\) is \(4\) units.