Answered

Uncover valuable information and solutions with IDNLearn.com's extensive Q&A platform. Get the information you need quickly and accurately with our reliable and thorough Q&A platform.

This circle is centered at the point [tex]\((3,2)\)[/tex], and the length of its radius is 5. What is the equation of the circle?

A. [tex]\((x^2 - 3) + (y^2 - 2) = 5^2\)[/tex]

B. [tex]\((x + 3)^2 + (y + 2)^2 = 5\)[/tex]

C. [tex]\((x - 2)^2 + (y - 3)^2 = 25\)[/tex]

D. [tex]\((x - 3)^2 + (y - 2)^2 = 25\)[/tex]

Sagot :

GinnyAnsweridn
To find the equation of a circle given its center and radius, we use the general formula for the equation of a circle:

[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] represents the radius.

Let's break down the components for the given circle:

1. The center of the circle is [tex]\((3, 2)\)[/tex], so [tex]\(h = 3\)[/tex] and [tex]\(k = 2\)[/tex].
2. The radius of the circle is [tex]\(5\)[/tex], so [tex]\(r = 5\)[/tex].

Substitute these values into the general formula:

[tex]\[ (x - 3)^2 + (y - 2)^2 = 5^2 \][/tex]

Next, simplify the right side of the equation:

[tex]\[ (x - 3)^2 + (y - 2)^2 = 25 \][/tex]

Now we just need to find the correct option that matches this equation. Let’s review the options:

A. [tex]\((x^2 - 3) + (y^2 - 2) = 5^2\)[/tex]

This is not correct because the equation is not properly formatted to match the standard circle equation. Also, it does not equal [tex]\(25\)[/tex] when simplified.

B. [tex]\((x + 3)^2 + (y + 2)^2 = 5\)[/tex]

This is not correct because the signs do not match the required [tex]\((x - 3)\)[/tex] and [tex]\((y - 2)\)[/tex], and the right side should be [tex]\(25\)[/tex], not [tex]\(5\)[/tex].

C. [tex]\((x - 2)^2 + (y - 3)^2 = 25\)[/tex]

This is not correct because the values for [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are swapped.

D. [tex]\((x - 3)^2 + (y - 2)^2 = 25\)[/tex]

This is correct because it matches our derived equation exactly.

Therefore, the correct answer is:

D. [tex]\((x - 3)^2 + (y - 2)^2 = 25\)[/tex]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.