Question

Please find the equation for the circle with a diameter whose endpoints are (2, -4) and (1,1). Write the standard equation for the circle. (Use integers or fractions for any numbers in the equation.)

A) (x - 1)² + (y + 1)² = 1
B) (x + 1)² + (y - 1)² = 1
C) (x - 2)² + (y + 2)² = 1
D) (x + 2)² + (y - 2)² = 1

Answer 1

The equation for the circle with a diameter whose endpoints are given is (x - 1.5)² + (y + 1.5)² = 6.5.

Step-by-step explanation:

To find the equation for a circle with a diameter whose endpoints are given, we can use the midpoint formula and the distance formula. The midpoint of the diameter is calculated as follows: (x, y) = ((x₁ + x₂)/2 , (y₁ + y₂)/2). Using the given endpoints (2, -4) and (1,1), we find the midpoint to be ((2+1)/2 , (-4+1)/2) = (1.5 , -1.5).

The radius of the circle is half the length of the diameter, so we need to find the distance between the midpoint and one of the endpoints. Using the distance formula, we have: d = √((x₂ - x₁)² + (y₂ - y₁)²). Plugging in the values, we get: d = √((1-1.5)² + (1-(-1.5))²) = √((-0.5)² + (2.5)²) = √(0.25 + 6.25) = √6.5.

Therefore, the equation of the circle with a diameter whose endpoints are (2, -4) and (1,1) is (x - 1.5)² + (y + 1.5)² = 6.5.