Question

a circle with center (4, 5) and radius r is shown on the graph. part a which equation is used to derive the equation of the circle? a. (a^{2}-b^{2}=r^{2}) b. (a^{2}+b^{2}=r^{2}) c. (x^{2}-y^{2}=r^{2}) d. (x^{2}+y^{2}=r^{2}) part b choose two expressions that can be substituted for the variables on the left - side of the equation to complete the derivation. a. (a - 4) b. (a + 4) c. (x - 4) d. (x + 4) e. (b - 5) f. (b + 5) g. (y - 5) h. (y + 5)

Explanation:

Step1: Recall the Pythagorean theorem

In the right - triangle formed in the circle diagram, by the Pythagorean theorem, the sum of the squares of the two legs of the right - triangle is equal to the square of the hypotenuse. Here, the lengths of the legs of the right - triangle are \(a\) and \(b\), and the hypotenuse is the radius \(r\). So \(a^{2}+b^{2}=r^{2}\).

Step2: Find the expressions for \(a\) and \(b\)

The horizontal distance \(a\) between the point \((x,y)\) and the center \((4,5)\) is \(x - 4\), and the vertical distance \(b\) between the point \((x,y)\) and the center \((4,5)\) is \(y - 5\).

Answer:

Part A: B. \(a^{2}+b^{2}=r^{2}\)
Part B: C. \(x - 4\), G. \(y - 5\)