Identify the equation of the circle that has its center at [tex]$(-16,30)$[/tex] and passes through a point.

A. [tex](x-16)^2+(y+30)^2=1256[/tex]
B. [tex](x+16)^2+(y-30)^2=1256[/tex]
C. [tex](x-16)^2+(y+30)^2=1156[/tex]
D. [tex](x+16)^2+(y-30)^2=1156[/tex]



Answer :

To find the equation of a circle with a given center and passing through a specific point, we follow these steps:

1. Identify the Center of the Circle: The center is given as [tex]\((h, k) = (-16, 30)\)[/tex].

2. Determine the Radius: We need to find the distance from the center to any point on the circle. Let's choose a point directly on the circle as [tex]\((x_1, y_1)\)[/tex]. If the circle passes through a point directly below the center by 20 units, our point would be at [tex]\((-16, 30 - 20) = (-16, 10)\)[/tex].

3. Distance Formula: The radius [tex]\(r\)[/tex] is the distance between the center [tex]\((-16, 30)\)[/tex] and the point [tex]\((-16, 10)\)[/tex]. We use the distance formula:
[tex]\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the points:
[tex]\[ r = \sqrt{(-16 - (-16))^2 + (10 - 30)^2} = \sqrt{0 + (-20)^2} = \sqrt{400} = 20 \][/tex]

4. Radius Squared: From the calculation, the radius [tex]\(r = 20\)[/tex], hence [tex]\(r^2 = 400\)[/tex].

5. Standard Form of Circle's Equation: The 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]
Substituting [tex]\(h = -16\)[/tex], [tex]\(k = 30\)[/tex], and [tex]\(r^2 = 400\)[/tex]:
[tex]\[ (x + 16)^2 + (y - 30)^2 = 400 \][/tex]

Comparing the obtained equation with the options provided:

A. [tex]\((x - 16)^2 + (y + 30)^2 = 1256\)[/tex]

B. [tex]\((x + 16)^2 + (y - 30)^2 = 1256\)[/tex]

C. [tex]\((x - 16)^2 + (y + 30)^2 = 1156\)[/tex]

D. [tex]\((x + 16)^2 + (y - 30)^2 = 1156\)[/tex]

The correct equation is:
[tex]\[ D: (x + 16)^2 + (y - 30)^2 = 400 \][/tex]

None of the given options exactly match this equation.

Hence if we have to choose option between that are the closest to our result, is:

[tex]\[ B: (x + 16)^2 + (y - 30)^2 which only differs in the radius squared \][/tex]