Question

write the standard equation for each of the circles in parts (a) through (e). 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 = 9). (type an equation. simplify your answer.) (b) the equation of the circle in standard form is (x^2 + y^2 =) . (type an equation. simplify your answer.)

Explanation:

Part (a)

Step1: Recall the standard form of a circle's equation.

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). For the circle \(x^2 + y^2 = 9\), we can rewrite it as \((x - 0)^2 + (y - 0)^2 = 3^2\). So the center is \((0, 0)\) and the radius \(r = 3\). But since the question here is just about the equation as given (maybe confirming the standard form), the equation is already in standard form \(x^2 + y^2 = 9\) (which is equivalent to \((x - 0)^2+(y - 0)^2 = 3^2\)).

Part (b)

Looking at the circle with center \((0,0)\) (from the diagram, the center is at the origin) and a point \((1,5)\)? Wait, no, maybe the other circle? Wait, the diagram has a circle with center \((0,0)\) and a point? Wait, no, the lower left circle has center \((0,0)\)? Wait, no, the lower right circle: let's check the points. The lower right circle has points \((0,3)\), \((3,0)\), \((0,-3)\), \((-3,0)\). Wait, no, the center is at \((0,0)\)? Wait, no, the center is at \((0,0)\) for the circle with points \((0,3)\), \((3,0)\) etc. Wait, the equation for a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2 = r^2\). If the center is \((0,0)\) and radius \(r\), then the equation is \(x^2 + y^2 = r^2\). But maybe the circle in part (b) has center \((0,0)\) and a point \((1,5)\)? No, that can't be. Wait, maybe the lower left circle: center \((0,0)\) and a point \((1,5)\)? No, that would make radius \(\sqrt{1^2 + 5^2}=\sqrt{26}\), so equation \(x^2 + y^2 = 26\). Wait, the box is \(x^2 + y^2=\), so we need to find \(r^2\). If the center is \((0,0)\) and the circle passes through \((1,5)\), then \(r^2=1^2 + 5^2=1 + 25 = 26\). So the equation is \(x^2 + y^2 = 26\).

Answer:

(for part b):
\(x^2 + y^2 = 26\)

(Note: For part (a), the equation is already given as \(x^2 + y^2 = 9\), which is the standard form with center \((0,0)\) and radius \(3\).)