Sure, let's solve this step-by-step!
1. Determine the center of the circle: The center is given as [tex]\((5, -4)\)[/tex]. In the equation of a circle [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], [tex]\(h\)[/tex] is the x-coordinate of the center, and [tex]\(k\)[/tex] is the y-coordinate of the center.
2. Determine the radius: To find the radius, we need to calculate the distance between the center [tex]\((5, -4)\)[/tex] and the point [tex]\((-3, 2)\)[/tex] using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}
\][/tex]
Here, [tex]\((x_1, y_1) = (5, -4)\)[/tex] and [tex]\((x_2, y_2) = (-3, 2)\)[/tex]. After performing the calculations, we find that the distance (radius) is 10. Therefore, [tex]\(r^2\)[/tex] is 100.
3. Substitute the center and radius into the circle's equation:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 100
\][/tex]
Thus, the equation of the circle is [tex]\((x - 5)^2 + (y + 4)^2 = 100\)[/tex].
Now we need to fit this in the form [tex]\( (x + \square )^2 + (y + \square )^2 = \square \)[/tex]:
- For [tex]\((x - 5)\)[/tex], we rewrite it as [tex]\((x + (-5))\)[/tex], so the first blank is [tex]\(-5\)[/tex].
- For [tex]\((y + 4)\)[/tex], it remains [tex]\((y + 4)\)[/tex], so the second blank is [tex]\(4\)[/tex].
- For [tex]\(100\)[/tex], it remains [tex]\(100\)[/tex], so the third blank is [tex]\(100\)[/tex].
Therefore, the complete answer is:
[tex]\((x + (-5))^2 + (y + 4)^2 = 100\)[/tex]
So, the correct answer is:
- Box 1: [tex]\(-5\)[/tex]
- Box 2: [tex]\(4\)[/tex]
- Box 3: [tex]\(100\)[/tex]
