A circle is centered at the point [tex](5,-4)[/tex] and passes through the point [tex](-3,2)[/tex].

The equation of this circle is:
[tex](x - 5)^2 + (y + 4)^2 = \square[/tex]

First, calculate the radius [tex]r[/tex] using the distance formula between the center [tex](5, -4)[/tex] and the point [tex](-3, 2)[/tex]:

[tex]r = \sqrt{(5 - (-3))^2 + (-4 - 2)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10[/tex]

Thus, the complete equation of the circle is:
[tex](x - 5)^2 + (y + 4)^2 = 100[/tex]



Answer :

To determine the equation of the circle, we need to follow these steps:

1. Identify the center of the circle: The center of the circle is given as [tex]\((5, -4)\)[/tex].

2. Identify a point on the circle: The circle passes through the point [tex]\((-3, 2)\)[/tex].

3. Calculate the radius of the circle:
The radius [tex]\(r\)[/tex] can be found using the distance formula between the center [tex]\((5, -4)\)[/tex] and the point on the circle [tex]\((-3, 2)\)[/tex]. However, in this case, it’s already provided that the radius squared ([tex]\(r^2\)[/tex]) is [tex]\(100.0\)[/tex].

4. Equation of the circle:
The general 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]
For the given circle, the center [tex]\((h, k)\)[/tex] is [tex]\((5, -4)\)[/tex], hence:
[tex]\[ (x - 5)^2 + (y + 4)^2 = 100 \][/tex]

To make the given equation fit the form [tex]\((x + \square)^2 + (y + \square)^2 = \square \)[/tex]:

[tex]\(\boxed{-5}\)[/tex] for [tex]\(h\)[/tex] (because [tex]\((x + (-5))\)[/tex] simplifies to [tex]\((x - 5)\)[/tex]),

[tex]\(\boxed{4}\)[/tex] for [tex]\(k\)[/tex] (because [tex]\((y + (4))\)[/tex] simplifies to [tex]\((y + 4)\)[/tex]),

[tex]\(\boxed{100.0}\)[/tex] for [tex]\(r^2\)[/tex].

So, the equation of the circle is:
[tex]\[ (x + (-5))^2 + (y + 4)^2 = 100.0 \][/tex]

If we transcribe this equation into the designated format, we get:
[tex]\[ (x + (-5))^2 + (y + 4)^2 = 100.0 \][/tex]

This can be rewritten as:
[tex]\[ (x - 5)^2 + (y + 4)^2 = 100.0 \][/tex]

Therefore, the answer is:
[tex]\[ (x + \boxed{-5})^2 + (y + \boxed{4})^2 = \boxed{100.0} \][/tex]