What is the equation of the circle with center [tex]\((3.2, -2.1)\)[/tex] and radius [tex]\(4.3\)[/tex]?

A. [tex]\((x + 2.1)^2 + (y - 3.2)^2 = 8.6\)[/tex]

B. [tex]\((x + 3.2)^2 + (y - 2.1)^2 = 4.3\)[/tex]

C. [tex]\((x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2\)[/tex]

D. [tex]\((x - 2.1)^2 - (y + 3.2)^2 = (4.3)^2\)[/tex]



Answer :

To determine the equation of the circle with a given center and radius, we 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 given by:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]

Given:
- Center [tex]\((h, k) = (3.2, -2.1)\)[/tex]
- Radius [tex]\(r = 4.3\)[/tex]

Substituting the given values into the standard form, we obtain:
[tex]\[ (x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2 \][/tex]

Now let's examine the given options to see which one matches this equation:

- Option A: [tex]\((x+2.1)^2+(y-3.2)^2=8.6\)[/tex]

This does not match our equation, as the signs of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are reversed, and the right-hand side [tex]\(8.6\)[/tex] does not equal [tex]\((4.3)^2\)[/tex].

- Option B: [tex]\((x+3.2)^2+(y-2.1)^2=4.3\)[/tex]

This does not match our equation, as the signs of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are reversed, and the right-hand side [tex]\(4.3\)[/tex] does not equal [tex]\((4.3)^2\)[/tex].

- Option C: [tex]\((x-3.2)^2+(y+2.1)^2=(4.3)^2\)[/tex]

This matches our equation exactly: [tex]\((x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2\)[/tex].

- Option D: [tex]\((x-2.1)^2-(y+3.2)^2=(4.3)^2\)[/tex]

This does not match our equation, as the values of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are incorrect, and there is a subtraction symbol in the equation, which is not part of the standard form for a circle.

Therefore, the correct equation of the circle is given in option C:
[tex]\[ (x-3.2)^2+(y+2.1)^2=(4.3)^2 \][/tex]

So the correct answer is:

Option C