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A circle is defined as the set of all points equidistant from a fixed center, formed from intersecting a plane with a cone. In Algebra, it can be represented as (x-a)² + (y-b)² = r², while in conic form, it's expressed as Ax² + Cy² + Dx + Ey + F = 0 with A = C. Key properties include the center (h,k), radius (r), and diameter. Degenerate cases occur when the radius is zero, reducing the circle to a point. Additionally, a circle has an eccentricity of zero, indicating it is perfectly circular.
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Conics: Circle By: Jay, Tyler, Bailey, Grace, Ryan, and Jesse
Origin of a Circle • A circle is formed when a plane is put through a cone and it is perpendicular to the axis of the cone and intersects each generator. • To construct a circle you need a fixed point (center), and a radius
Forms of a Circle • Algebra II form: (x-a)² + (y-b)² = r² • Conic Form: Ax²+Cy²+Dx+Ey+F=0 if A=C • Rotated Form: Ax²+Bxy+Cy²+Dx+Ey+F=0 if B²-4AC<0 Standard Form: x²+y²=r² (at origin) (x-h)²+(y-k)²=r² (not at origin) Relationships – Center (a,b) or (h,k), radius= r Radius- Half the width, Diameter- The width
Circle info • Degenerate Case – A circle with a radius of zero is a point, radius approaches 0, until it becomes a point. • Application – Changing the radius or origin to move the circle. • Eccentricity- The eccentricity of a circle is zero. Eccentricity is how much the conic deviates from being circular.
