Question

1.3 homework question 14, 1.3.63 part 2 of 3 hw score: 81.9%, 13 of 21 points points: 0 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 ( x^2 + y^2 = 49 ). (type an equation. simplify your answer.) (b) the equation of the circle in standard form is ( (x - 5)^2 + y^2 = 4 ). (type an equation. simplify your answer.)

Explanation:

Part (a)

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: Identify center and radius

From the graph, the center is at \((0, 0)\) (since it's at the origin, as seen from the symmetric points \((-7,0)\), \((7,0)\), \((0,7)\), \((0,-7)\)). The radius can be found by the distance from the center to a point on the circle, e.g., from \((0,0)\) to \((7,0)\) is \(7\), so \(r = 7\), and \(r^2 = 49\). Wait, but the given answer in the problem was \(x^2 + y^2 = 49\) (maybe a typo in the original problem's shown answer as \(40\) was incorrect). Let's correct: using center \((0,0)\) and \(r = 7\), the equation is \(x^2 + y^2 = 7^2 = 49\).

Step3: Write the equation

Substituting \(h = 0\), \(k = 0\), \(r = 7\) into the standard form: \((x - 0)^2 + (y - 0)^2 = 7^2\), which simplifies to \(x^2 + y^2 = 49\).

Part (b)

Step1: Recall the standard circle equation

The standard form is \((x - h)^2 + (y - k)^2 = r^2\), with \((h, k)\) as center and \(r\) as radius.

Step2: Identify center and radius

From the graph, the center is \((5, 0)\) (given as "Center" at \((5,0)\)). A point on the circle is \((3, 0)\). The radius \(r\) is the distance between \((5, 0)\) and \((3, 0)\), which is \(|5 - 3| = 2\), so \(r^2 = 4\).

Step3: Write the equation

Substituting \(h = 5\), \(k = 0\), \(r = 2\) into the standard form: \((x - 5)^2 + (y - 0)^2 = 2^2\), which simplifies to \((x - 5)^2 + y^2 = 4\) (matches the given answer in the problem, so this is correct).

Part (c) (assuming there was a part (c) not fully shown, but let's check the first two parts' corrections)

Answer:

(for part a correction):
\(x^2 + y^2 = 49\)