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The equation [tex]$(x-5)^2 + (y+3)^2 = r^2$[/tex] represents circle [tex]$T$[/tex]. The point [tex]$S(-1, 6)$[/tex] lies on the circle. What is [tex][tex]$r$[/tex][/tex], the length of the radius of circle [tex]$T$[/tex]?

A. [tex]\sqrt{12}[/tex]
B. [tex]\sqrt{15}[/tex]
C. [tex]3\sqrt{5}[/tex]
D. [tex]3\sqrt{13}[/tex]

Sagot :

To determine the radius [tex]\(r\)[/tex] of the circle described by the equation [tex]\((x-5)^2 + (y+3)^2 = r^2\)[/tex], we need to find the distance between the center of the circle [tex]\(T\)[/tex] and the point [tex]\(S\)[/tex] that lies on the circle.

First, identify the coordinates:
- The center of the circle [tex]\(T\)[/tex] is at [tex]\((5, -3)\)[/tex].
- The point [tex]\(S\)[/tex] is at [tex]\((-1, 6)\)[/tex].

To find the distance between these two points, use the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substitute [tex]\( (x_1, y_1) = (5, -3) \)[/tex] and [tex]\( (x_2, y_2) = (-1, 6) \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{((-1) - 5)^2 + (6 - (-3))^2} \][/tex]
Simplify inside the parentheses:
[tex]\[ = \sqrt{(-6)^2 + (9)^2} \][/tex]
Square the values:
[tex]\[ = \sqrt{36 + 81} \][/tex]
Add the squared values:
[tex]\[ = \sqrt{117} \][/tex]
Simplify the square root:
[tex]\[ = \sqrt{9 \times 13} = 3 \sqrt{13} \][/tex]

Thus, the radius [tex]\(r\)[/tex] of the circle is [tex]\(3 \sqrt{13}\)[/tex].

So, the length of the radius of circle [tex]\(T\)[/tex] is:
[tex]\[ \boxed{3 \sqrt{13}} \][/tex]
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