Question

fudzisako · Accepted Answer

To find the slope of the line that passes through the centers of the two circles, we first need to determine the coordinates of the centers.
Given that AB is a diameter of Circle 1, the center of Circle 1, denoted as

O_{1}

, will be the midpoint of AB. Similarly, since BC is a diameter of Circle 2, the center of Circle 2, denoted as

O_{2}

, will be the midpoint of BC.
Let's find the coordinates of the centers:
The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) are given by:

(\frac{x ₁ + x ₂}{2}, \frac{y ₁ + y ₂}{2})

For Circle 1, the coordinates of A and B are (1, 1) and (5, 3) respectively. Using the midpoint formula, we can find

O_{1}

:

O_{1} = (\frac{1 + 5}{2}, \frac{1 + 3}{2}) = (3, 2)

For Circle 2, the coordinates of B and C are (5, 3) and (5, 9) respectively. Applying the midpoint formula, we can find

O_{2}

:

O_{2} = (\frac{5 + 5}{2}, \frac{3 + 9}{2}) = (5, 6)

Now that we have the coordinates of the two centers,

O_{1} = (3, 2)

and

O_{2} = (5, 6)

, we can find the slope of the line passing through these two points using the slope formula:
The slope between two points (x₁, y₁) and (x₂, y₂) is given by:

m = \frac{y ₂ - y ₁}{x ₂ - x ₁}

Substituting the coordinates of the two centers into the formula, we get:

m = \frac{6 - 2}{5 - 3} = \frac{4}{2} = 2

Therefore, the slope of the line passing through the centers of the two circles is 2.

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