Question

The equation of the circle with extremities \( (1, 3) \) and \( (5, 7) \) of the diameter is:

• A. \( x^2 + y^2 + 6x + 10y + 26 = 0 \)
• B. \( x^2 + y^2 - 6x - 10y + 26 = 0 \)
• C. \( x^2 + y^2 - 6x + 10y + 26 = 0 \)
• D. \( x^2 + y^2 - 6x - 10y - 26 = 0 \)

Hint:

To find the equation of a circle given the endpoints of the diameter, calculate the midpoint to get the center and the distance between the center and either endpoint to get the radius.

Answer 1

The Correct Option is B

The general equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center and \( r \) is the radius. The center of the circle is the midpoint of the diameter. The extremities of the diameter are given as \( (1, 3) \) and \( (5, 7) \). Step 1: Find the center of the circle.
The center is the midpoint of the diameter, so we calculate the midpoint of \( (1, 3) \) and \( (5, 7) \) using the formula for the midpoint: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the values: \[ \left( \frac{1 + 5}{2}, \frac{3 + 7}{2} \right) = (3, 5) \] So, the center of the circle is \( (3, 5) \). Step 2: Find the radius of the circle.
The radius is the distance from the center \( (3, 5) \) to either endpoint of the diameter. Using the distance formula, we calculate the distance between \( (3, 5) \) and \( (1, 3) \): \[ r = \sqrt{(3 - 1)^2 + (5 - 3)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \] Thus, the radius is \( 2\sqrt{2} \). Step 3: Write the equation of the circle.
The general equation of the circle is: \[ (x - 3)^2 + (y - 5)^2 = (2\sqrt{2})^2 \] Simplifying: \[ (x - 3)^2 + (y - 5)^2 = 8 \] Expanding the equation: \[ (x^2 - 6x + 9) + (y^2 - 10y + 25) = 8 \] \[ x^2 + y^2 - 6x - 10y + 34 = 8 \] \[ x^2 + y^2 - 6x - 10y + 26 = 0 \] Thus, the equation of the circle is \( x^2 + y^2 - 6x - 10y + 26 = 0 \).