To write the equations into circle standard form, we need to expand the squared terms and group the variables accordingly:
1). (x-1)²+(y+3)²=4
Expanding the squared terms:
x² - 2x + 1 + y² + 6y + 9 = 4
Rearranging:
x² + y² - 2x + 6y + 6 = 0
Completing the square for x and y:
(x - 1)² + (y + 3)² = 2²
Standard form:
(x - 1)² + (y + 3)² = 4
2.) (x-2)²+(y+1)²=16
Expanding the squared terms:
x² - 4x + 4 + y² + 2y + 1 = 16
Rearranging:
x² + y² - 4x + 2y - 11 = 0
Completing the square for x and y:
(x - 2)² + (y + 1)² = 4²
Standard form:
(x - 2)² + (y + 1)² = 16
3.) (x-1)²+(y+4)²=9
Expanding the squared terms:
x² - 2x + 1 + y² + 8y + 16 = 9
Rearranging:
x² + y² - 2x + 8y + 8 = 0
Completing the square for x and y:
(x - 1)² + (y + 4)² = 3²
Standard form:
(x - 1)² + (y + 4)² = 9
4.) x²+(y-3)²=14
Rearranging:
x² + y² - 6y + 9 = 14
Completing the square for y:
x² + (y - 3)² = 5²
Standard form:
x² + (y - 3)² = 25
For a circle with center at point (h, k) and radius r, the distance formula to find the distance between the center and any point (x, y) on the circle is:
d = √[(x - h)^2 + (y - k)^2]
For the first problem, we have the center at (13, -13) and a radius of 4. To find the equation of the circle, we can substitute these values into the distance formula:
d = √[(x - 13)^2 + (y + 13)^2] = 4
Squaring both sides and simplifying, we get:
(x - 13)^2 + (y + 13)^2 = 16
This is the equation of the circle.
For the second problem, we have the center at (-13, -16) and a point on the circle at (-10, -16). Again, we can use the distance formula to find the radius of the circle:
d = √[(-10 + 13)^2 + (-16 + 16)^2] = 3
So the radius is 3. To find the equation of the circle, we can substitute the values of the center and radius into the distance formula:
d = √[(x + 13)^2 + (y + 16)^2] = 3
Squaring both sides and simplifying, we get:
(x + 13)^2 + (y + 16)^2 = 9
This is the equation of the circle.
For the third problem, we have the center at (3, -2) and a radius of 4. Using the same distance formula, we can find the equation of the circle:
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Answer:
To write the equations into circle standard form, we need to expand the squared terms and group the variables accordingly:
1). (x-1)²+(y+3)²=4
Expanding the squared terms:
x² - 2x + 1 + y² + 6y + 9 = 4
Rearranging:
x² + y² - 2x + 6y + 6 = 0
Completing the square for x and y:
(x - 1)² + (y + 3)² = 2²
Standard form:
(x - 1)² + (y + 3)² = 4
2.) (x-2)²+(y+1)²=16
Expanding the squared terms:
x² - 4x + 4 + y² + 2y + 1 = 16
Rearranging:
x² + y² - 4x + 2y - 11 = 0
Completing the square for x and y:
(x - 2)² + (y + 1)² = 4²
Standard form:
(x - 2)² + (y + 1)² = 16
3.) (x-1)²+(y+4)²=9
Expanding the squared terms:
x² - 2x + 1 + y² + 8y + 16 = 9
Rearranging:
x² + y² - 2x + 8y + 8 = 0
Completing the square for x and y:
(x - 1)² + (y + 4)² = 3²
Standard form:
(x - 1)² + (y + 4)² = 9
4.) x²+(y-3)²=14
Rearranging:
x² + y² - 6y + 9 = 14
Completing the square for y:
x² + (y - 3)² = 5²
Standard form:
x² + (y - 3)² = 25
For a circle with center at point (h, k) and radius r, the distance formula to find the distance between the center and any point (x, y) on the circle is:
d = √[(x - h)^2 + (y - k)^2]
For the first problem, we have the center at (13, -13) and a radius of 4. To find the equation of the circle, we can substitute these values into the distance formula:
d = √[(x - 13)^2 + (y + 13)^2] = 4
Squaring both sides and simplifying, we get:
(x - 13)^2 + (y + 13)^2 = 16
This is the equation of the circle.
For the second problem, we have the center at (-13, -16) and a point on the circle at (-10, -16). Again, we can use the distance formula to find the radius of the circle:
d = √[(-10 + 13)^2 + (-16 + 16)^2] = 3
So the radius is 3. To find the equation of the circle, we can substitute the values of the center and radius into the distance formula:
d = √[(x + 13)^2 + (y + 16)^2] = 3
Squaring both sides and simplifying, we get:
(x + 13)^2 + (y + 16)^2 = 9
This is the equation of the circle.
For the third problem, we have the center at (3, -2) and a radius of 4. Using the same distance formula, we can find the equation of the circle:
d = √[(x - 3)^2 + (y + 2)^2] = 4
Squaring both sides and simplifying, we get:
(x - 3)^2 + (y + 2)^2 = 16
This is the equation of the circle