Find the equations of the circles for which
the points given below are the end points
of a diameter.
`(1,2), (4,6)`

Answer

To find the equation of the circle for which the points (1, 2) and (4, 6) are the endpoints of the diameter, we can follow these steps:

### Step 1: Identify the endpoints
Let the endpoints of the diameter be:
- Point 1: \( (x_1, y_1) = (1, 2) \)
- Point 2: \( (x_2, y_2) = (4, 6) \)

### Step 2: Use the equation of the circle
The equation of a circle with endpoints of the diameter at \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]

### Step 3: Substitute the points into the equation
Substituting the values of \( x_1, y_1, x_2, y_2 \):
\[ (x - 1)(x - 4) + (y - 2)(y - 6) = 0 \]

### Step 4: Expand the equation
Now, we will expand both products:
1. Expand \( (x - 1)(x - 4) \):
\[ (x - 1)(x - 4) = x^2 - 4x - x + 4 = x^2 - 5x + 4 \]

2. Expand \( (y - 2)(y - 6) \):
\[ (y - 2)(y - 6) = y^2 - 6y - 2y + 12 = y^2 - 8y + 12 \]

### Step 5: Combine the expanded terms
Now, combine the expanded terms:
\[ x^2 - 5x + 4 + y^2 - 8y + 12 = 0 \]

### Step 6: Simplify the equation
Combine like terms:
\[ x^2 + y^2 - 5x - 8y + 16 = 0 \]

### Final Equation
Thus, the equation of the circle is:
\[ x^2 + y^2 - 5x - 8y + 16 = 0 \]

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