Question

work: section 1.3 homework question 14, 1.3.63 part 1 of 3 hw score: 63.49%, 13.33 of 21 points points: 0.33 of 1 write the standard equation for each of the circles in parts (a) through (c). the coordinates of the center and the radius for each circle are integers. (a) the equation of the circle in standard form is (type an equation. simplify your answer.)

Explanation:

Step1: Identify center and radius

The center of the circle is at the origin \((0,0)\) (since it's the intersection of the axes of symmetry). The radius can be found by the distance from the center to a point on the circle, e.g., to \((4,0)\), so radius \(r = 4\).

Step2: Use 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. Substituting \(h = 0\), \(k = 0\), and \(r = 4\), we get \((x - 0)^2 + (y - 0)^2 = 4^2\).

Step3: Simplify the equation

Simplifying, we have \(x^2 + y^2 = 16\).

Answer:

\(x^2 + y^2 = 16\)