Question

find the standard equation of the circle passing through a given point with a given center. center (10,4) and passing through (0, -3) the equation in standard form is \boxed{}. (simplify your answer.)

Explanation:

Step1: Recall the standard circle equation

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: Find the radius

The radius \(r\) is the distance between the center \((10,4)\) and the point \((0,-3)\) on the circle. Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(r=\sqrt{(0 - 10)^2+(-3 - 4)^2}=\sqrt{(- 10)^2+(-7)^2}=\sqrt{100 + 49}=\sqrt{149}\).

Step3: Write the circle equation

Substitute \(h = 10\), \(k = 4\), and \(r^2=149\) into the standard equation: \((x - 10)^2+(y - 4)^2 = 149\).

Answer:

\((x - 10)^2+(y - 4)^2 = 149\)