Question

does the point $(-1,sqrt{15})$ lie on the circle centered at the origin and containing the point $(0,4)$? explain.
yes, because the equation for the circle is $x^{2}+y^{2}=4$ and $(-1)^{2}+(sqrt{15})^{2}=4$.
no, because the equation for the circle is $x^{2}+y^{2}=4$ and $(-1)^{2}+(sqrt{15})^{2}
eq4$.
no, because the equation for the circle is $x^{2}+y^{2}=16$ and $(-1)^{2}+(sqrt{15})^{2}
eq16$.
yes, because the equation for the circle is $x^{2}+y^{2}=16$ and $(-1)^{2}+(sqrt{15})^{2}=16$.

Explanation:

Step1: Find circle radius

The circle is centered at the origin and contains (0,4). Radius $r = \sqrt{(0-0)^2 + (4-0)^2} = 4$. Equation: $x^2 + y^2 = 4^2 = 16$.

Step2: Check the point

Substitute $(-1, \sqrt{15})$ into the equation: $(-1)^2 + (\sqrt{15})^2 = 1 + 15 = 16$.

Answer:

Yes, because the equation for the circle is $x^2 + y^2 = 16$ and $(-1)^2 + (\sqrt{15})^2 = 16$.