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Questions and Answers
What is the standard form of the equation of a circle centered at (4, -1) with a radius of 7?
What is the standard form of the equation of a circle centered at (4, -1) with a radius of 7?
- (x + 4)^2 + (y - 1)^2 = 49
- (x - 4)^2 + (y - 1)^2 = 7
- (x - 4)^2 + (y + 1)^2 = 49 (correct)
- (x + 4)^2 + (y + 1)^2 = 7
What is the center of the circle given by the equation x^2 + y^2 - 6x - 10y + 18 = 0?
What is the center of the circle given by the equation x^2 + y^2 - 6x - 10y + 18 = 0?
- (6, 10)
- (-3, -5)
- (3, 5) (correct)
- (4, -2)
What is the radius of the circle represented by the equation (x - 3)^2 + (y - 5)^2 = 16?
What is the radius of the circle represented by the equation (x - 3)^2 + (y - 5)^2 = 16?
- 2
- 8
- 16
- 4 (correct)
Which of the following forms indicates that an equation represents a circle?
Which of the following forms indicates that an equation represents a circle?
What is the first step in converting the equation x + y + 6x - 7 = 0 into standard form?
What is the first step in converting the equation x + y + 6x - 7 = 0 into standard form?
What is the center of the circle described in the equations provided?
What is the center of the circle described in the equations provided?
What is the radius of the circle that has the center (-2, 2)?
What is the radius of the circle that has the center (-2, 2)?
Which equation correctly represents the circle with radius 4 and centered at (0, 0)?
Which equation correctly represents the circle with radius 4 and centered at (0, 0)?
Which form is used to express the circle equation in the form (x-h)² + (y-k)² = r²?
Which form is used to express the circle equation in the form (x-h)² + (y-k)² = r²?
In which quadrant would the center (-2, 2) of the circle lie?
In which quadrant would the center (-2, 2) of the circle lie?
If the equation of the circle is (x + 3)² + (y - 1)² = r², what is the y-coordinate of the center?
If the equation of the circle is (x + 3)² + (y - 1)² = r², what is the y-coordinate of the center?
How would you represent the radius of a circle in the equation format (x - h)² + (y - k)² = ?
How would you represent the radius of a circle in the equation format (x - h)² + (y - k)² = ?
What is the value of r² in the equation of the circle (x - 2)² + (y + 1)² = r² when r = 6?
What is the value of r² in the equation of the circle (x - 2)² + (y + 1)² = r² when r = 6?
What is the center of the circle represented by the equation $(x + 5)^2 + (y - 2)^2 = 49$?
What is the center of the circle represented by the equation $(x + 5)^2 + (y - 2)^2 = 49$?
What is the radius of the circle defined by the equation $(x + 5)^2 + (y - 2)^2 = 49$?
What is the radius of the circle defined by the equation $(x + 5)^2 + (y - 2)^2 = 49$?
Using the distance formula, what is the distance between the points P1(-5, 3) and P2(7, 11)?
Using the distance formula, what is the distance between the points P1(-5, 3) and P2(7, 11)?
What is the midpoint between the points P1(-5, 3) and P2(7, 11)?
What is the midpoint between the points P1(-5, 3) and P2(7, 11)?
What is the slope of the line that passes through the points P1(2, 3) and P2(5, 11)?
What is the slope of the line that passes through the points P1(2, 3) and P2(5, 11)?
Which of the following equations represents a circle with center (2, -1) and radius 6?
Which of the following equations represents a circle with center (2, -1) and radius 6?
From which form can the equation of a circle be derived?
From which form can the equation of a circle be derived?
If a circle has a radius of 7 and is defined by the equation $(x + 6)^2 + (y - 1)^2 = 49$, what is the center of the circle?
If a circle has a radius of 7 and is defined by the equation $(x + 6)^2 + (y - 1)^2 = 49$, what is the center of the circle?
Which of the following statements is true regarding the slope of vertical lines?
Which of the following statements is true regarding the slope of vertical lines?
What is the general form of the equation of a circle centered at (h, k)?
What is the general form of the equation of a circle centered at (h, k)?
Which equation represents a circle centered at the origin with a radius of 4?
Which equation represents a circle centered at the origin with a radius of 4?
What is the correct equation of the circle with a center at (0, 3) and a radius of 5?
What is the correct equation of the circle with a center at (0, 3) and a radius of 5?
Given the equation of the circle $x^2 + (y + 2)^2 = 36$, what is the center of the circle?
Given the equation of the circle $x^2 + (y + 2)^2 = 36$, what is the center of the circle?
If a circle has the equation $(x - 1)^2 + (y + 4)^2 = 49$, what is its radius?
If a circle has the equation $(x - 1)^2 + (y + 4)^2 = 49$, what is its radius?
What is the standard equation of the circle whose center is (1, 7) and radius is $2 \sqrt{13}$?
What is the standard equation of the circle whose center is (1, 7) and radius is $2 \sqrt{13}$?
Determine the coordinate point that is 6 km away from the station located at (0, -4).
Determine the coordinate point that is 6 km away from the station located at (0, -4).
Which equation represents the possible locations of an epicenter 1 km from the shore?
Which equation represents the possible locations of an epicenter 1 km from the shore?
What are the possible coordinates of the epicenter if it is 1 km away from the shore?
What are the possible coordinates of the epicenter if it is 1 km away from the shore?
How does the distance formula relate to determining the radius of a circle in the context of an earthquake epicenter?
How does the distance formula relate to determining the radius of a circle in the context of an earthquake epicenter?
What does the center of a circle represent in relation to the epicenter of an earthquake?
What does the center of a circle represent in relation to the epicenter of an earthquake?
What is the radius of the circle used to denote the possible epicenter locations?
What is the radius of the circle used to denote the possible epicenter locations?
What is the significance of the endpoint coordinates (-5, 3) and (7, 11) in determining the radius and center of the circle?
What is the significance of the endpoint coordinates (-5, 3) and (7, 11) in determining the radius and center of the circle?
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Study Notes
Circles Basics
- A circle consists of all points equidistant from a fixed center point, with distance referred to as the radius.
- The standard equation format for a circle centered at ((h, k)) is ((x - h)^2 + (y - k)^2 = r^2).
- For a circle centered at the origin, the equation simplifies to (x^2 + y^2 = r^2).
Example Circle Equations
-
Circle with center at (0, 0) and radius 5:
(x^2 + y^2 = 25). -
Circle centered at (0, 3) with radius 6:
((x - 0)^2 + (y - 3)^2 = 36) simplifies to (x^2 + (y - 3)^2 = 36).
Finding the Standard Form
- To convert the equation of a circle into standard form, complete the square for (x) and (y).
- Example: For the equation (x^2 - 8x + y^2 + 2y - 32 = 0), complete the square to yield: ((x - 4)^2 + (y + 1)^2 = 49).
Center and Radius Extraction
-
Given the equation (x^2 + y^2 - 6x - 10y + 18 = 0):
- Complete the square to find center at (3, 5) and radius as 4 units.
-
For (4x^2 + 4y^2 + 12x - 4y - 90 = 0):
- Center is at ((-2, 2)) with a radius of 5 units.
Graphing Circles
- A circle can be graphed using its center and radius on a Cartesian plane.
- Example: For center (2, -1) and radius 6, graph the equation ((x - 2)^2 + (y + 1)^2 = 36).
Distance and Slope Formulas
-
The distance formula between two points ((x_1, y_1)) and ((x_2, y_2)) is calculated as: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).
-
The slope of a line through points (P_1(x_1, y_1)) and (P_2(x_2, y_2)) is defined as: (m = \frac{y_2 - y_1}{x_2 - x_1}).
Problem Solving Applications
- For a circle determined by the endpoints of a diameter, calculate the center and radius using the midpoint and distance formulas.
- An example involving seismic data can be described through the equation of circles related to distance from a given location.
Conic Sections
- Circles are one type of conic section formed by the intersection of a plane and a cone.
- Their shapes are defined by the angle at which the cone is sliced, creating different conic sections like ellipses, parabolas, and hyperbolas.
Summary of Methods
- Completing the square, extracting the center and radius, and graphing circles are fundamental methods in understanding and applying circle equations in problems related to geometry and real-world scenarios.
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