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Select the correct answer.

Circle C has a center at [tex]$(-2,10)$[/tex] and contains the point [tex]$P(10,5)$[/tex]. Which equation represents circle C?

A. [tex]$(x-2)^2+(y+10)^2=13$[/tex]

B. [tex]$(x-2)^2+(y+10)^2=169$[/tex]

C. [tex]$(x+2)^2+(y-10)^2=13$[/tex]

D. [tex]$(x+2)^2+(y-10)^2=169$[/tex]

Sagot :

GinnyAnswer
To determine the equation of circle [tex]\( C \)[/tex] that has a center at [tex]\((-2, 10)\)[/tex] and contains the point [tex]\(P(10, 5)\)[/tex], follow these steps:

1. Identify the center and point on the circle:
- Center: [tex]\((-2, 10)\)[/tex]
- Point on the circle: [tex]\(P(10, 5)\)[/tex]

2. Find the radius of the circle using the distance formula between the center [tex]\((-2, 10)\)[/tex] and the point [tex]\(P(10, 5)\)[/tex]:

[tex]\[ \text{Radius} = \sqrt{(10 - (-2))^2 + (5 - 10)^2} \][/tex]

[tex]\[ = \sqrt{(10 + 2)^2 + (5 - 10)^2} \][/tex]

[tex]\[ = \sqrt{12^2 + (-5)^2} \][/tex]

[tex]\[ = \sqrt{144 + 25} \][/tex]

[tex]\[ = \sqrt{169} \][/tex]

[tex]\[ = 13 \][/tex]

3. Use the standard form of the equation of a circle:
The standard form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is

[tex]\[ (x - h)^2 + (y - k)^2 = r^2. \][/tex]

Here, the center [tex]\((h, k)\)[/tex] is [tex]\((-2, 10)\)[/tex] and the radius [tex]\(r\)[/tex] is [tex]\(13\)[/tex].

4. Plug the center and radius into the standard form:
- Center: [tex]\((-2, 10)\)[/tex]
- Radius: [tex]\(13\)[/tex]

[tex]\[ (x - (-2))^2 + (y - 10)^2 = 13^2 \][/tex]

[tex]\[ (x + 2)^2 + (y - 10)^2 = 169 \][/tex]

So, the correct equation representing circle [tex]\( C \)[/tex] is:

[tex]\[ \boxed{(x+2)^2+(y-10)^2=169} \][/tex]

Thus, the correct answer is D.
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