Equation Of Circle

For a circle with center and radius , the equation is: We are given the center . To find the radius, use the distance formula between the center and the point : Thus, the equation of the circle is: Expanding: Thus, the equation of the circle is .

We are given the equation . To find the center and radius, we need to rewrite this in the standard form of a circle equation . Step 1: Divide through by 2 Step 2: Complete the square For -terms: For -terms: Step 3: Substitute back and simplify Substituting into the equation: Simplifying: Thus, the center is and the radius is .

The equation of a circle with center and radius is given by: Given the center , we know the circle touches the X-axis. The radius is equal to the y-coordinate of the center, which is . Substitute into the standard circle equation: Expanding and simplifying: Thus, the equation of the circle is .

The general equation of a circle is: where is the center of the circle and is its radius. Given that the circle touches the x-axis, the radius of the circle is equal to the y-coordinate of its center, i.e., . Since the center lies on the line , we have: We are also given that the circle passes through the point . Substituting this point into the equation of the circle, we get: Now, substitute and into the above equation: Simplifying: Expanding: Solving for : Using the quadratic formula: So, or . Since the circle is in the first quadrant, we take , and therefore . Now, the equation of the circle is: Expanding: Thus, the correct equation is: Hence, the correct answer is (B).