To find the equation for a circle given its center and a point it passes through, we need to follow these steps:
1. Identify the center and the point on the circle:
- The center of the circle is [tex]\((4, 3)\)[/tex].
- The circle passes through the point [tex]\((3, -4)\)[/tex].
2. Calculate the radius:
- The radius of the circle can be calculated using the distance formula between the center [tex]\((h, k) = (4, 3)\)[/tex] and the point [tex]\((x_1, y_1) = (3, -4)\)[/tex]:
[tex]\[
\text{Radius} = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\][/tex]
Plugging in the numbers:
[tex]\[
\text{Radius} = \sqrt{(3 - 4)^2 + (-4 - 3)^2} = \sqrt{(-1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50}
\][/tex]
3. Write the general form of the circle's equation:
- The general equation for a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Using our specific values where [tex]\(h = 4\)[/tex], [tex]\(k = 3\)[/tex], and [tex]\(r = \sqrt{50}\)[/tex], we substitute into the formula:
[tex]\[
(x - 4)^2 + (y - 3)^2 = (\sqrt{50})^2
\][/tex]
4. Simplify the equation:
- Squaring the radius [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
(\sqrt{50})^2 = 50
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 4)^2 + (y - 3)^2 = 50
\][/tex]
So, the equation for the circle with center [tex]\((4, 3)\)[/tex] and passing through [tex]\((3, -4)\)[/tex] is:
[tex]\[
(x - 4)^2 + (y - 3)^2 = 50
\][/tex]
