Question

4
type the correct answer in each box.
a circle is centered at the point (-3, 2) and passes through the point (1, 5). the radius of the circle is units. the point (-7, ) lies on this circle.
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Explanation:

Step1: Calculate the radius using distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, the center of the circle is $(x_1,y_1)=(-3,2)$ and a point on the circle is $(x_2,y_2)=(1,5)$. So, $r=\sqrt{(1 - (-3))^2+(5 - 2)^2}=\sqrt{(4)^2+(3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$.

Step2: Find the y - coordinate of the point on the circle

The equation of a circle with center $(h,k)=(-3,2)$ and radius $r = 5$ is $(x+3)^2+(y - 2)^2=25$. Substitute $x=-7$ into the equation: $(-7 + 3)^2+(y - 2)^2=25$, which simplifies to $(-4)^2+(y - 2)^2=25$, then $16+(y - 2)^2=25$, $(y - 2)^2=25-16 = 9$, $y-2=\pm3$. So $y=2 + 3=5$ or $y=2-3=-1$.

Answer:

5; 5 or -1