Questions: Centro en C(2,-1) y cuya circunferencia pasa por el punto P(2,4).

Solution Steps

To find the equation of a circle given its center \( C(2, -1) \) and a point \( P(2, 4) \) on the circumference, we can use the standard form of the circle equation \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. First, we calculate the radius \(r\) as the distance between the center and the point on the circumference using the distance formula. Then, we substitute the center coordinates and the radius into the circle equation.

Given:

• Center \( C(2, -1) \)
• Point on the circumference \( P(2, 4) \)

The radius \( r \) is the distance between the center \( C(2, -1) \) and the point \( P(2, 4) \). Using the distance formula: \[ r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} \] Substituting the given values: \[ r = \sqrt{(2 - 2)^2 + (4 - (-1))^2} = \sqrt{0 + 5^2} = \sqrt{25} = 5 \]

The standard form of the equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substituting \( h = 2 \), \( k = -1 \), and \( r = 5 \): \[ (x - 2)^2 + (y - (-1))^2 = 5^2 \] Simplifying: \[ (x - 2)^2 + (y + 1)^2 = 25 \]

Final Answer