Question

1. geometry *

in the standard (x,y) coordinate plane, how many points are both 5 coordinate units from the origin and also 2 coordinate units from the line y = 0?
f. 0
g. 1
h. 2
j. 4

Explanation:

Step1: Find the equation of points 5 units from origin

The equation of a circle with center at the origin $(0,0)$ and radius $r = 5$ is $x^{2}+y^{2}=r^{2}=25$ according to the standard - form of the equation of a circle $(x - a)^{2}+(y - b)^{2}=r^{2}$, where $(a,b)=(0,0)$.

Step2: Find the possible y - values for points 2 units from $y = 0$

The line $y = 0$ is the x - axis. Points that are 2 coordinate units from the line $y = 0$ have $y=2$ or $y = - 2$.

Step3: Substitute y - values into the circle equation

When $y = 2$, we substitute into $x^{2}+y^{2}=25$. So $x^{2}+2^{2}=25$, which gives $x^{2}=25 - 4=21$, and $x=\pm\sqrt{21}$.
When $y=-2$, we substitute into $x^{2}+y^{2}=25$. So $x^{2}+(-2)^{2}=25$, which gives $x^{2}=25 - 4 = 21$, and $x=\pm\sqrt{21}$.

The four points are $(\sqrt{21},2),(-\sqrt{21},2),(\sqrt{21}, - 2),(-\sqrt{21},-2)$.

Answer:

J. 4