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Questions and Answers
What is the general form of the equation of a circle with center at the origin and radius 2?
What is the general form of the equation of a circle with center at the origin and radius 2?
- x^2 + y^2 = 4 (correct)
- x^2 + y^2 = 2
- x^2 + y^2 = 8
- x^2 + y^2 - 4 = 0 (correct)
Identify the center and radius of the circle represented by the equation x^2 + y^2 - 8x + 7 = 0.
Identify the center and radius of the circle represented by the equation x^2 + y^2 - 8x + 7 = 0.
- Center (4, 0), Radius 3 (correct)
- Center (2, 3), Radius 5
- Center (3, -1), Radius 2
- Center (4, 0), Radius 4
What transformation is applied to rewrite the equation of a circle in general form?
What transformation is applied to rewrite the equation of a circle in general form?
- Distributing
- Completing the square (correct)
- Factoring
- Substitution
Given the equation 4x^2 + 4y^2 + 8x - 8y – 4 = 0, what is the radius when converted to standard form?
Given the equation 4x^2 + 4y^2 + 8x - 8y – 4 = 0, what is the radius when converted to standard form?
What is the correct general form for the circle centered at (-1, 2) with a radius of 3?
What is the correct general form for the circle centered at (-1, 2) with a radius of 3?
Which architectural structures can exemplify the use of conic sections?
Which architectural structures can exemplify the use of conic sections?
In the equation of a circle x^2 + 2x + y^2 - 4y + 2 = 0, what operation is used to determine the center?
In the equation of a circle x^2 + 2x + y^2 - 4y + 2 = 0, what operation is used to determine the center?
What is the form of a circle with center not at the origin?
What is the form of a circle with center not at the origin?
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Study Notes
Conic Sections and Architecture
- Conic sections are mathematical shapes formed by intersecting a cone and a plane.
- The shapes (circles, ellipses, parabolas, and hyperbolas) have applications in architecture, which can be seen in famous structures like the Sydney Harbor Bridge and the Canton Tower.
Circles: Applications in Architecture
- They are used in the design of domes and arches.
- The circular shape provides structural strength and stability.
Equation of a Circle:
- Standard Form: (x – h)2 + (y - k)2 = r2
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
- General Form: Ax2 + By2 + Cx + Dy + E = 0
- This is a more complex form of the equation of a circle and can be converted to the standard form to get the center and radius.
Finding the Center and Radius of a Circle from its General Equation
- Completing the square is the process for solving for the center and radius of a circle given its general form.
- Example: x2 + y2 -8x + 7 = 0
- Rearrange the equation: x2 -8x + y2 = -7
- Complete the square: (x2 – 8x + 16) + y2 = -7 + 16
- Simplify: (x - 4 )2 + y2 = 9
- Determine the center: C = (4, 0)
- Determine the radius: r=3
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