Question

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a circle is centered at m(0, 0). the point b(-4, \sqrt{5}) is on the circle.
where does the point a(5, -1) lie?
choose 1 answer:
a inside the circle
b on the circle
c outside the circle

Explanation:

Step1: Calculate the radius of the circle

The distance between the center $M(0,0)$ and the point $B(-4,\sqrt{5})$ on the circle is the radius $r$. Using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, we have $r=\sqrt{(-4 - 0)^2+(\sqrt{5}-0)^2}=\sqrt{16 + 5}=\sqrt{21}$.

Step2: Calculate the distance between the center and point $A$

Calculate the distance $d_{MA}$ between $M(0,0)$ and $A(5,-1)$ using the distance formula. $d_{MA}=\sqrt{(5 - 0)^2+(-1 - 0)^2}=\sqrt{25+1}=\sqrt{26}$.

Step3: Compare the distances

Since $\sqrt{26}>\sqrt{21}$ (i.e., $d_{MA}>r$), point $A$ is outside the circle.

Answer:

C. Outside the circle