In the standard (x,y) coordinate plane, the circle centerd at (1,3) that passes through (4,7) is the set of all points that are:

Options

A. 5 coordinate units from (1,3)
B. 5 coordinate units from both (1,3) and (4,7)
C. 5 coordinate units from the line segment with endpoints (1,3) and (4,7)
D. equidistant from (1,3) and (4,7)

Answer

To solve the problem, we need to find the equation of the circle centered at (1, 3) that passes through the point (4, 7). Here are the steps to derive the solution:

### Step 1: Identify the center of the circle
The center of the circle is given as (1, 3).

### Step 2: Find the radius of the circle
To find the radius, we need to calculate the distance between the center (1, 3) and the point (4, 7) using the distance formula:
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \((x_1, y_1) = (1, 3)\) and \((x_2, y_2) = (4, 7)\).

Substituting the values:
\[ r = \sqrt{(4 - 1)^2 + (7 - 3)^2} \]
\[ = \sqrt{(3)^2 + (4)^2} \]
\[ = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \]
So, the radius \(r\) is 5 units.

### Step 3: Write the equation of the circle
The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 1\), \(k = 3\), and \(r = 5\):
\[ (x - 1)^2 + (y - 3)^2 = 5^2 \]
\[ (x - 1)^2 + (y - 3)^2 = 25 \]

### Step 4: Interpret the result
The equation \((x - 1)^2 + (y - 3)^2 = 25\) represents all points \((x, y)\) that are 5 units away from the center (1, 3). Therefore, the set of all points that satisfy this equation forms a circle with a radius of 5 centered at (1, 3).

### Final Answer
The circle centered at (1, 3) that passes through (4, 7) is the set of all points that are 5 coordinate units from (1, 3).

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