Work each of the following. Find the equation of a circle with center at (-4, 3), passing through the point (5, 8).Write it in center-radius form.

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Recall that the center-radius form of a circle's equation is given by $\left(x - h\right)^2 + \left(y - k\right)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Identify the center $(h, k)$ from the problem: here, the center is $(-4, 3)$, so $h = -4$ and $k = 3$.

Calculate the radius $r$ by finding the distance between the center $(-4, 3)$ and the point on the circle $(5, 8)$ using the distance formula: $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Substitute the coordinates into the distance formula: $r = \sqrt{(5 - (-4))^2 + (8 - 3)^2}$, then simplify inside the square root.

Write the equation of the circle by plugging $h$, $k$, and $r^2$ into the center-radius form: $\left(x - (-4)\right)^2 + \left(y - 3\right)^2 = r^2$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation of a Circle in Center-Radius Form

The center-radius form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. This form directly relates the circle's geometric properties to its algebraic equation.

Distance Formula

The distance formula calculates the distance between two points (x1, y1) and (x2, y2) as √[(x2 - x1)^2 + (y2 - y1)^2]. It is used here to find the radius by measuring the distance from the center to a point on the circle.

Substitution into the Circle Equation

After finding the radius, substitute the center coordinates and radius squared into the center-radius form. This step finalizes the equation of the circle, ensuring it passes through the given point.

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