Get expert insights and reliable answers to your questions on IDNLearn.com. Discover detailed answers to your questions with our extensive database of expert knowledge.

Question 14 of 25

The equation below describes a circle. What are the coordinates of the center of the circle?

[tex]\[ (x-6)^2+(y+5)^2=15^2 \][/tex]

A. [tex]\((-6,-5)\)[/tex]

B. [tex]\((-6,5)\)[/tex]

C. [tex]\((6,5)\)[/tex]

D. [tex]\((6,-5)\)[/tex]


Sagot :

To find the coordinates of the center of the circle given the equation [tex]\((x - 6)^2 + (y + 5)^2 = 15^2\)[/tex], we need to recognize the standard form of a circle's equation. The general form of the equation of a circle is:

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

Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] is the radius.

For the given equation [tex]\((x - 6)^2 + (y + 5)^2 = 15^2\)[/tex], we can identify [tex]\(h\)[/tex] and [tex]\(k\)[/tex] by comparing it with the standard form. Let's break it down:

1. [tex]\((x - 6)^2\)[/tex] corresponds to the term [tex]\((x - h)^2\)[/tex]. Thus, [tex]\(h = 6\)[/tex].
2. [tex]\((y + 5)^2\)[/tex] can be rewritten as [tex]\((y - (-5))^2\)[/tex], which aligns with [tex]\((y - k)^2\)[/tex]. Therefore, [tex]\(k = -5\)[/tex].

From this, we determine that the center of the circle [tex]\((h, k)\)[/tex] is at the coordinates [tex]\((6, -5)\)[/tex].

Hence, the correct answer is:
D. [tex]\((6, -5)\)[/tex]