1) Find the center and radius of a circle with the following equations:
a) \(x^{2} + y^{2} = 64\)
b) \(x^{2} + y^{2} = 121\)
c) \(x^{2} + y^{2} = 9\)
d) \(x^{2} + y^{2} = 5\)
2) Sketch a graph of each circle in #1.
3) The standard form of a circle is \((x - h)^{2} + (y - k)^{2} = r^{2}\) where (h, k) is the location of the center and \(r\) is the length of the radius. Find the center and radius of each circle.
a) \((x - 3)^{2} + (y - 2)^{2} = 16\)
b) \((x + 4)^{2} + (y - 1)^{2} = 81\)
c) \(\left(x + \dfrac{1}{2}\right)^{2} + \left(y - \dfrac{3}{4}\right)^{2} = \dfrac{25}{49}\)
d) \(x^{2} + (y + 3.2)^{2} = 1.44\)
4) Sketch a graph of the circles in #3a & #3b.
5) To obtain an equation in standard form, we sometimes need to form perfect square trinomials. We can do this by adding a special amount to "complete the square." In the following expressions, fill in the blank that completes the square (and therefore makes a perfect square trinomial).
a) \(x^{2} + 8x + \) ______ .
b) \(y^{2} - 6y + \) ______ .
c) \(x^{2} - 9x + \) ______ .
d) \(y^{2} - \dfrac{5}{8}y + \) ______ .
e) \(x^{2} + 1.6x + \) _____ .
f) \(y^{2} + 2y + \)_____ .
6) Write the binomial being squared from problem #5, the perfect square trinomials. The first one is done for you.
a) (answer) \((x + 4)^{2}\)
7) Fill in the blanks
a) \(x^{2} + y^{2} = 64\)
b) \(x^{2} + y^{2} = 121\)
c) \(x^{2} + y^{2} = 9\)
d) \(x^{2} + y^{2} = 5\)
2) Sketch a graph of each circle in #1.
3) The standard form of a circle is \((x - h)^{2} + (y - k)^{2} = r^{2}\) where (h, k) is the location of the center and \(r\) is the length of the radius. Find the center and radius of each circle.
a) \((x - 3)^{2} + (y - 2)^{2} = 16\)
b) \((x + 4)^{2} + (y - 1)^{2} = 81\)
c) \(\left(x + \dfrac{1}{2}\right)^{2} + \left(y - \dfrac{3}{4}\right)^{2} = \dfrac{25}{49}\)
d) \(x^{2} + (y + 3.2)^{2} = 1.44\)
4) Sketch a graph of the circles in #3a & #3b.
5) To obtain an equation in standard form, we sometimes need to form perfect square trinomials. We can do this by adding a special amount to "complete the square." In the following expressions, fill in the blank that completes the square (and therefore makes a perfect square trinomial).
a) \(x^{2} + 8x + \) ______ .
b) \(y^{2} - 6y + \) ______ .
c) \(x^{2} - 9x + \) ______ .
d) \(y^{2} - \dfrac{5}{8}y + \) ______ .
e) \(x^{2} + 1.6x + \) _____ .
f) \(y^{2} + 2y + \)_____ .
6) Write the binomial being squared from problem #5, the perfect square trinomials. The first one is done for you.
a) (answer) \((x + 4)^{2}\)
7) Fill in the blanks
b) Now, write the binomials being squared. Your equation should be in standard form.
c) Finally, name the center and radius of the circle.
8) Write the equation in standard form and find the location of the center and length of the radius.
a) \(x^{2} - 8x + y^{2} + 2y = 8\)
b) \(x^{2} + 14x + y^{2} - 10y - 7 = 0\)
c) \(x^{2} + y^{2} + 20x + 16y - 32 = 0\)
d) \(x^{2} - 6x + y^{2} + 5y = \dfrac{3}{4}\)
9) Write an equation given its center and length of its radius
a) Center \((0, 0)\), Radius = \(3\)
b) Center \((0, 4)\), Radius = \(5\)
c) Center \((-2, -1)\), Radius = \(12\)
d)Center \((5, -6)\), Radius = \(\dfrac{4}{5}\)
e) Center \((-3.1, 6.3)\), Radius = \(2.1\)
f) Center \(\left(0, \dfrac{5}{6}\right)\), Radius = \(\dfrac{7}{9}\)
10) Suppose a circle has a center at \((4, -5)\) and a radius length of \(3\).
a) Write the equation of the circle.
b) Determine if \((2, -7)\) is inside, on, or outside the circle. Use Desmos if necessary.
11) Give the equation of a circle with the following information.
a) Center \((0, 0)\) and point on the circle \((0, 3)\)
b) Center \((0, 3)\) and point on the circle \((5, 3)\)
c) Center \((-2, -1)\) and point on the circle \((2, 2)\)
d) The two points that represent the endpoints of the diameter of the circle \((-3, 3)\) and \((-3, -5)\)
e) The two points that represent the endpoints of the diameter of the circle \((5, 4)\) and \((-3, -2)\)
f) The two points that represent the endpoints of the diameter of the circle \((5, -6)\) and \((1, -2)\)
12) A cellular phone company places towers in three locations to service a local town. The company superimposes a coordinate grid over a map of the town. The first cellular tower has a range of 5 hectometers and is located at \((3, -2)\), the second one has a range of 6 hectometers and is located at \((-5, 5)\) and the third one has a range of \(7\) hectometers and is located at \((7, 7)\).
Name an integer ordered pair which would not receive service (would not be within the circles). Use Desmos if necessary.
13) A circle with equation \((x + 3)^{2} + (y - 7)^{2} = 25\) has a chord with endpoints at \((1, 10)\) and \((1, 4)\). Write the equation of another circle with a radius length of \(5\) and has the same chord. Use Desmos if necessary.
c) Finally, name the center and radius of the circle.
8) Write the equation in standard form and find the location of the center and length of the radius.
a) \(x^{2} - 8x + y^{2} + 2y = 8\)
b) \(x^{2} + 14x + y^{2} - 10y - 7 = 0\)
c) \(x^{2} + y^{2} + 20x + 16y - 32 = 0\)
d) \(x^{2} - 6x + y^{2} + 5y = \dfrac{3}{4}\)
9) Write an equation given its center and length of its radius
a) Center \((0, 0)\), Radius = \(3\)
b) Center \((0, 4)\), Radius = \(5\)
c) Center \((-2, -1)\), Radius = \(12\)
d)Center \((5, -6)\), Radius = \(\dfrac{4}{5}\)
e) Center \((-3.1, 6.3)\), Radius = \(2.1\)
f) Center \(\left(0, \dfrac{5}{6}\right)\), Radius = \(\dfrac{7}{9}\)
10) Suppose a circle has a center at \((4, -5)\) and a radius length of \(3\).
a) Write the equation of the circle.
b) Determine if \((2, -7)\) is inside, on, or outside the circle. Use Desmos if necessary.
11) Give the equation of a circle with the following information.
a) Center \((0, 0)\) and point on the circle \((0, 3)\)
b) Center \((0, 3)\) and point on the circle \((5, 3)\)
c) Center \((-2, -1)\) and point on the circle \((2, 2)\)
d) The two points that represent the endpoints of the diameter of the circle \((-3, 3)\) and \((-3, -5)\)
e) The two points that represent the endpoints of the diameter of the circle \((5, 4)\) and \((-3, -2)\)
f) The two points that represent the endpoints of the diameter of the circle \((5, -6)\) and \((1, -2)\)
12) A cellular phone company places towers in three locations to service a local town. The company superimposes a coordinate grid over a map of the town. The first cellular tower has a range of 5 hectometers and is located at \((3, -2)\), the second one has a range of 6 hectometers and is located at \((-5, 5)\) and the third one has a range of \(7\) hectometers and is located at \((7, 7)\).
Name an integer ordered pair which would not receive service (would not be within the circles). Use Desmos if necessary.
13) A circle with equation \((x + 3)^{2} + (y - 7)^{2} = 25\) has a chord with endpoints at \((1, 10)\) and \((1, 4)\). Write the equation of another circle with a radius length of \(5\) and has the same chord. Use Desmos if necessary.
14) Find the equation of each graphed circle below.