Circles Basics and Equations

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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?

(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?

(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?

4 (correct)

Which of the following forms indicates that an equation represents a circle?

x^2 + y^2 = r^2 Signup and view all the answers

What is the first step in converting the equation x + y + 6x - 7 = 0 into standard form?

Combine like terms Signup and view all the answers

What is the center of the circle described in the equations provided?

(-2, 2) Signup and view all the answers

What is the radius of the circle that has the center (-2, 2)?

5 Signup and view all the answers

Which equation correctly represents the circle with radius 4 and centered at (0, 0)?

16 Signup and view all the answers

Which form is used to express the circle equation in the form (x-h)² + (y-k)² = r²?

25 Signup and view all the answers

In which quadrant would the center (-2, 2) of the circle lie?

2 Signup and view all the answers

If the equation of the circle is (x + 3)² + (y - 1)² = r², what is the y-coordinate of the center?

1 Signup and view all the answers

How would you represent the radius of a circle in the equation format (x - h)² + (y - k)² = ?

5 Signup and view all the answers

What is the value of r² in the equation of the circle (x - 2)² + (y + 1)² = r² when r = 6?

36 Signup and view all the answers

What is the center of the circle represented by the equation $(x + 5)^2 + (y - 2)^2 = 49$?

(-5, 2) Signup and view all the answers

What is the radius of the circle defined by the equation $(x + 5)^2 + (y - 2)^2 = 49$?

7 Signup and view all the answers

Using the distance formula, what is the distance between the points P1(-5, 3) and P2(7, 11)?

$14$ Signup and view all the answers

What is the midpoint between the points P1(-5, 3) and P2(7, 11)?

(1, 7) Signup and view all the answers

What is the slope of the line that passes through the points P1(2, 3) and P2(5, 11)?

$\frac{8}{3}$ Signup and view all the answers

Which of the following equations represents a circle with center (2, -1) and radius 6?

$(x - 2)^2 + (y + 1)^2 = 36$ Signup and view all the answers

From which form can the equation of a circle be derived?

Standard form Signup and view all the answers

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?

(-6, 1) Signup and view all the answers

Which of the following statements is true regarding the slope of vertical lines?

Their slope is always undefined. Signup and view all the answers

What is the general form of the equation of a circle centered at (h, k)?

$(x - h)^2 + (y - k)^2 = r^2$ Signup and view all the answers

Which equation represents a circle centered at the origin with a radius of 4?

$x^2 + y^2 = 16$ Signup and view all the answers

What is the correct equation of the circle with a center at (0, 3) and a radius of 5?

$x^2 + (y - 3)^2 = 25$ Signup and view all the answers

Given the equation of the circle $x^2 + (y + 2)^2 = 36$, what is the center of the circle?

(0, -2) Signup and view all the answers

If a circle has the equation $(x - 1)^2 + (y + 4)^2 = 49$, what is its radius?

7 Signup and view all the answers

What is the standard equation of the circle whose center is (1, 7) and radius is $2 \sqrt{13}$?

$(x - 1)^2 + (y - 7)^2 = 52$ Signup and view all the answers

Determine the coordinate point that is 6 km away from the station located at (0, -4).

(±6, -4) Signup and view all the answers

Which equation represents the possible locations of an epicenter 1 km from the shore?

$x^2 + (y + 4)^2 = 36$ Signup and view all the answers

What are the possible coordinates of the epicenter if it is 1 km away from the shore?

(±3.31, 1), (±5.2, -1) Signup and view all the answers

How does the distance formula relate to determining the radius of a circle in the context of an earthquake epicenter?

It generates the radius used in the standard circle equation. Signup and view all the answers

What does the center of a circle represent in relation to the epicenter of an earthquake?

The starting point of the seismic activity. Signup and view all the answers

What is the radius of the circle used to denote the possible epicenter locations?

6 km Signup and view all the answers

What is the significance of the endpoint coordinates (-5, 3) and (7, 11) in determining the radius and center of the circle?

They are part of the diameter used to locate the center. Signup and view all the answers

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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Description

This quiz covers the fundamental concepts of circles, including their definition, standard equations, and methods for finding center and radius. You'll learn to convert circle equations into standard form and extract key information from given equations.