Question

write the equation in standard form for the circle passing through (-1, 0) centered at the origin.

Explanation:

Step1: Recall circle standard form

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 center coordinates

The center is at the origin, so \(h = 0\) and \(k = 0\).

Step3: Calculate the radius

The circle passes through \((-1, 0)\). The radius \(r\) is the distance from the center \((0,0)\) to the point \((-1, 0)\). Using the distance formula \(r=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we substitute \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(-1,0)\): \(r=\sqrt{(-1 - 0)^2+(0 - 0)^2}=\sqrt{1 + 0}=1\).

Step4: Substitute into standard form

Substitute \(h = 0\), \(k = 0\), and \(r = 1\) into \((x - h)^2+(y - k)^2=r^2\): \((x - 0)^2+(y - 0)^2=1^2\), which simplifies to \(x^2 + y^2 = 1\).

Answer:

\(x^2 + y^2 = 1\)