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Problem 629

# A circle whose center is at $$\mathrm{C}(-4,2)$$ passes through the point $$\mathrm{D}(-3,5)$$. Find $$R$$, the length of the radius, in radical form.

Expert verified
The length of the radius, R, of the circle in radical form is: $$R = \sqrt{10}$$
See the step by step solution

## Step 1: Identify the coordinates of the points C and D

We have the given coordinates of the center of the circle, point C, and a point on the circle, point D: C(-4, 2) and D(-3, 5). Our goal is to find the radial length (R).

## Step 2: Distance formula

The distance formula allows us to find the distance between two points in the Cartesian plane. The formula is given by: $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Where D is the distance, (x1, y1) are the coordinates of the first point, and (x2, y2) are the coordinates of the second point.

## Step 3: Apply the distance formula

To find the length of the radius (R), we will calculate the distance between points C and D using the distance formula, which will be the same as the magnitude of the radius vector. $$R = \sqrt{((-3) - (-4))^2 + (5 - 2)^2}$$

## Step 4: Compute values and simplify

Now plug the values into the distance formula and simplify: $$R = \sqrt{(1)^2 + (3)^2}$$ $$R = \sqrt{1 + 9}$$ $$R = \sqrt{10}$$

The length of the radius, R, of the circle in radical form is: $$R = \sqrt{10}$$

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