Circle and Its Equations

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Questions and Answers

What is the standard form of a circle's equation?

(x + h)^2 + (y + k)^2 = r^2

(x - h)^2 + (y - k)^2 = r^2 (correct)

Ax^2 + By^2 + Dx + Ey + F = 0

x^2 + y^2 = r

Given the equation (x + 1)^2 + (y - 2)^2 = 9, what is the radius?

3 (correct)

What is the center of the circle represented by the equation (x - 4)^2 + (y - 7)^2 = 25?

(7, 4)

(-4, 7)

(4, 7) (correct)

(4, -7)

In the general equation Ax^2 + Cy^2 + Dx + Ey + F = 0, what do A and C represent?

Quadratic terms Signup and view all the answers

If the center of a circle is at (0, 0) and it has a radius of 6, what is the correct equation?

x^2 + y^2 = 36 Signup and view all the answers

What is the significance of the focus in relation to a parabola?

It is the fixed point located inside the curve. Signup and view all the answers

How is the length of the latus rectum related to the focal distance?

It is always 4 times the focal distance. Signup and view all the answers

What does the eccentricity of a parabola always equal?

1 Signup and view all the answers

What is the role of the axis of the parabola?

It divides the parabola into two equal parts. Signup and view all the answers

If the focal distance (p) is positive for a horizontal parabola, which direction does it open?

To the right Signup and view all the answers

What is the general form of a conic section represented by the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$?

General conic section Signup and view all the answers

Which of the following represents a circle in the conic discriminant classification?

A > 0, B = 0, C > 0 Signup and view all the answers

In the conic discriminant, what does the expression $B^2 - 4AC$ indicate?

Nature of the conic section Signup and view all the answers

Which of the following equations represents a parabola based on the conic discriminant?

x^2 - 4y = 0 Signup and view all the answers

What characterizes the distance from the center to any point on a circle?

It is constant and called the radius. Signup and view all the answers

For the conic section represented by $9x - 4y - 18x - 16y - 43 = 0$, what type of conic does it represent?

Not a conic section Signup and view all the answers

What does it mean when $B^2 - 4AC < 0$ in a conic section's equation?

The conic is an ellipse. Signup and view all the answers

Which expression corresponds to a condition that confirms a conic is a hyperbola?

$B^2 - 4AC > 0$ Signup and view all the answers

What does the term 'quadratic terms' refer to in the context of conic sections?

Terms that have an exponent of two on their variable. Signup and view all the answers

What are the coordinates of the center for the circle defined by the equation $(x - 3)^2 + y^2 = 16$?

(3, 0) Signup and view all the answers

Which equation represents a circle with a center at (-11, 8) and a radius of 8?

$(x + 11)^2 + (y - 8)^2 = 64$ Signup and view all the answers

What is the radius of the circle represented by the equation $(x - 2)^2 + (y - 5)^2 = 64$?

8 Signup and view all the answers

Which of the following is the correct standard form of the equation of a circle centered at (0,0) with a radius of 2?

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

What is the center of the circle represented by the equation $(x - 2)^2 + (y + 4)^2 = 9$?

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

From which equation can you derive that the radius of the circle is 3?

$(x - 5)^2 + (y + 2)^2 = 9$ Signup and view all the answers

Which equation corresponds to a circle with a center at (5, -4) and a radius of 3?

$(x - 5)^2 + (y + 4)^2 = 9$ Signup and view all the answers

What is the standard form of the equation of a circle with a radius of 4 and centered at (3, 0)?

$(x - 3)^2 + y^2 = 16$ Signup and view all the answers

What is the general equation of a circle with center (-7, -5) and radius 9?

(x + 7)² + (y + 5)² = 81 Signup and view all the answers

Which point is the vertex of a parabola defined in the context?

The midpoint of the curve Signup and view all the answers

How is the directrix related to a parabola?

It lies outside and parallel to the parabola curve. Signup and view all the answers

Given the parameters h=6 and k=11, what is the general equation of the circle with center (6, 11) and radius 10?

(x - 6)² + (y - 11)² = 100 Signup and view all the answers

What characteristic does a parabola possess regarding its points?

Points are equidistant from the vertex and the directrix Signup and view all the answers

What is the radius of the circle defined with center (-7, -5) when its general equation is (x + 7)² + (y + 5)² = r²?

9 Signup and view all the answers

What aspects define the shape and position of the parabola?

The vertex and the directrix line Signup and view all the answers

In the context of the circle equation, what do the variables h and k represent?

The coordinates of the center Signup and view all the answers

If the radius of a circle is increased, which statement is likely true?

The diameter of the circle increases. Signup and view all the answers

What will happen to the graph of a parabola if the vertex is moved vertically?

The entire parabola will shift up or down. Signup and view all the answers

Study Notes

Circle

A circle consists of all points that are equidistant from a fixed point known as the center.
The distance from the center to any point on the circle is referred to as the radius.
The standard form of the circle's equation is ((x - h)^2 + (y - k)^2 = r^2), where ((h, k)) are the coordinates of the center and (r) is the radius.

Circle Equation Derivation

For a circle with center at ((-7, -5)) and radius (9):
- Equation: ((x + 7)^2 + (y + 5)^2 = 9^2) simplifies to ((x + 7)^2 + (y + 5)^2 = 81).
A second circle with center ((6, 11)) and radius (10):
- Equation: ((x - 6)^2 + (y - 11)^2 = 10^2), yielding: ((x - 6)^2 + (y - 11)^2 = 100).

General Circle Equation

General form of the circle equation is (Ax^2 + Ay^2 + Dx + Ey + F = 0), with (A) representing quadratic terms, (D) and (E) linear terms, and (F) a constant.

Conic Sections

Conic sections include two notable curves: circles and parabolas.
A parabola is a curve where each point is equidistant from a fixed point (focus) and a given line (directrix).
Parabolas resemble elongated semi-circles and have distinct characteristics.

Parts of a Parabola

Vertex (V): The midpoint of the parabola curve also serves as the midpoint between the focus and directrix.
Directrix (DL): A line that is outside and parallel to the parabola.
Focus (F): The fixed point from which distances to the directrix are measured.
Focal Distance (a): Distance from the focus to vertex, also reflects relationship to directrix.
Latus Rectum (LR): Chord through the focus, parallel to the directrix, where its length is always four times the focal distance.

Properties of Parabolas

Axis of Symmetry: A line that divides the parabola into two equal parts, passing through the vertex and focus, perpendicular to the directrix.
Eccentricity (e): Ratio of distances from a point on the curve to the focus and directrix line; for parabolas, (e = 1).

Parabola Standard Forms

For a parabola with vertex at the origin:
- Horizontal: (y = 4px) opens to the right if (p > 0).
- Vertical: (x = 4py) opens upward if (p > 0).

Discriminant for Conic Sections

The conic discriminant (B^2 - 4AC) determines the type of conic represented by the equation:
- Positive indicates an ellipse or circle; zero denotes a parabola; negative signifies a hyperbola.

Circle Properties in Examples

Example circles identified with equations show centers and radii:
- Radius calculations involve solving from standard and general forms, ensuring understanding of fundamental properties and relations.

Graphing Circles and Parabolas

Understanding equations enables graphing of circles with specific characteristics such as center and radius.
Parabolas can be graphed using vertex and focus parameters, facilitating visual representations of their unique shapes.

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Description

This quiz focuses on the properties of circles, including their equations and derivation. You'll learn about the standard and general forms of circle equations and explore conic sections. Test your understanding of key concepts and calculations related to circles.