Question

1. which is an equation of a circle with center (2, -1) that passes through the point (3, 4)?

○ ((x + 2)^2 + (y - 1)^2 = 26)
○ ((x - 2)^2 + (y + 1)^2 = 13)
○ ((x + 2)^2 + (y - 1)^2 = 13)
○ ((x - 2)^2 + (y + 1)^2 = 26)

Explanation:

Step1: Recall the circle equation formula

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step2: Identify the center coordinates

Given center \((2, -1)\), so \(h = 2\), \(k = -1\). Substitute into the formula: \((x - 2)^2 + (y - (-1))^2 = r^2\), which simplifies to \((x - 2)^2 + (y + 1)^2 = r^2\).

Step3: Calculate the radius

The circle passes through \((3, 4)\). The radius \(r\) is the distance between \((2, -1)\) and \((3, 4)\). Using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), we have \(r = \sqrt{(3 - 2)^2 + (4 - (-1))^2} = \sqrt{1^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26}\). Then \(r^2 = (\sqrt{26})^2 = 26\).

Step4: Form the circle equation

Substitute \(r^2 = 26\) into \((x - 2)^2 + (y + 1)^2 = r^2\), we get \((x - 2)^2 + (y + 1)^2 = 26\).

Answer:

D. \((x - 2)^2 + (y + 1)^2 = 26\)