Question

find the center and radius for each equation of a circle.

1. ((x + 5)^2 + (y - 6)^2 = 121) 2. ((x - 3)^2 + (y + 9)^2 = 36)

write the equation for the circle shown in each graph.

1. image of a circle on a grid with center a 4. image of a circle on a grid with center b

write the equation for each circle with the given radius and center.

1. radius: 2, center: ((4, -3)) 6. radius: 7, center: ((-1, 10))

determine whether each given point lies on the circle with the given radius and center.

1. ((6, 8)); radius: 4, center: ((3, 5)) 8. ((sqrt{17}, 4)); radius: 9, center: ((0, -4))

graph each equation.

1. ((x - 5)^2 + (y - 1)^2 = 49) 10. (x^2 + (y - 3)^2 = 25) grids for graphing
2. if the center of a circle is at ((a, b)) and the radius of the circle is (c), how would you write the equation of the circle?
3. a machine part is a 4-in.-square piece of metal plate with a round hole 2 in. in diameter drilled through the center. what is the equation of the circle given that the piece of metal is placed in quadrant 1 with its corner at the origin?

Explanation:

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: Analyze equation 1: \((x + 5)^2 + (y - 6)^2 = 121\)

Rewrite \(x + 5\) as \(x - (-5)\). So, comparing with \((x - h)^2 + (y - k)^2 = r^2\), we have \(h = -5\), \(k = 6\), and \(r^2 = 121\). Taking the square root, \(r = 11\) (since radius is positive).

Step3: Analyze equation 2: \((x - 3)^2 + (y + 9)^2 = 36\)

Rewrite \(y + 9\) as \(y - (-9)\). Then, \(h = 3\), \(k = -9\), and \(r^2 = 36\), so \(r = 6\).

Step4: For problem 3 (graph):

First, find the center of circle A from the graph. Looking at the grid, the center seems to be at \((-4, -3)\) (by counting grid units). The radius: from center to a point on the circle, say horizontally or vertically. Counting, the radius is 3 (since from \(x=-4\) to \(x=-1\) is 3 units). So equation is \((x + 4)^2 + (y + 3)^2 = 9\) (since \(r^2 = 3^2 = 9\)).

Step5: For problem 4 (graph):

Center of circle B is at \((3, 2)\) (from grid). Radius: from center to a point on the circle, e.g., to \((6, 2)\) is 3 units? Wait, no, looking at the graph, from center \((3,2)\) to the top (3,6) is 4 units? Wait, let's check again. The circle goes from y= -2 to y=6, so diameter is 8, radius 4. Wait, center at (3,2), radius 4. So equation: \((x - 3)^2 + (y - 2)^2 = 16\) (since \(4^2 = 16\)).

Step6: Problem 5: radius 2, center (4, -3)

Using standard form: \((x - 4)^2 + (y + 3)^2 = 2^2 = 4\).

Step7: Problem 6: radius 7, center (-1, 10)

Equation: \((x + 1)^2 + (y - 10)^2 = 7^2 = 49\).

Step8: Problem 7: Check if (6,8) is on circle with radius 4, center (3,5)

First, calculate the distance between (6,8) and (3,5). Using distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). So \(d = \sqrt{(6 - 3)^2 + (8 - 5)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24\), which is more than radius 4. So the point is not on the circle.

Step9: Problem 8: Check if \((\sqrt{17}, 4)\) is on circle with radius 9, center (0, -4)

Distance between \((\sqrt{17}, 4)\) and (0, -4): \(d = \sqrt{(\sqrt{17} - 0)^2 + (4 - (-4))^2} = \sqrt{17 + 64} = \sqrt{81} = 9\), which equals the radius. So the point is on the circle.

Step10: Problem 9: Graph \((x - 5)^2 + (y - 1)^2 = 49\)

Center at (5,1), radius 7 (since \(7^2 = 49\)). To graph, plot center (5,1), then draw a circle with radius 7 (count 7 units in all directions from center).

Step11: Problem 10: Graph \(x^2 + (y - 3)^2 = 25\)

Center at (0,3), radius 5 (since \(5^2 = 25\)). Plot center (0,3), draw circle with radius 5.

Step12: Problem 11: General equation

If center is (a, b) and radius c, then the equation is \((x - a)^2 + (y - b)^2 = c^2\).

Step13: Problem 12: Machine part

The square is 4-in square, corner at origin in Quadrant 1. So the square has corners at (0,0), (4,0), (4,4), (0,4). The center of the square (and thus the circle, since hole is through center) is at (2,2). The hole has diameter 2 in, so radius 1 in. So equation of the circle: \((x - 2)^2 + (y - 2)^2 = 1^2 = 1\).

Answer:

s for each sub - problem:

1. Center: \((-5, 6)\), Radius: \(11\)
2. Center: \((3, -9)\), Radius: \(6\)
3. Equation: \(\boldsymbol{(x + 4)^2 + (y + 3)^2 = 9}\)
4. Equation: \(\boldsymbol{(x - 3)^2 + (y - 2)^2 = 16}\)
5. Equation: \(\boldsymbol{(x - 4)^2 + (y + 3)^2 = 4}\)
6. Equation: \(\boldsymbol{(x + 1)^2 + (y - 10)^2 = 49}\)
7. The point \((6, 8)\) \(\boldsymbol{\text{does not lie}}\) on the circle.
8. The point \((\sqrt{17}, 4)\) \(\boldsymbol{\text{lies}}\) on the circle.
9. Graph: Center \((5,1)\), radius \(7\) (plot and draw circle).
10. Graph: Center \((0,3)\), radius \(5\) (plot and draw circle).
11. Equation: \(\boldsymbol{(x - a)^2 + (y - b)^2 = c^2}\)
12. Equation: \(\boldsymbol{(x - 2)^2 + (y - 2)^2 = 1}\)