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Conic Sections

Conic Sections. Circle. Conics. Four Basic Conic Shapes: Circle Parabola Ellipse Hyperbola A conic section (or simply conic ) is the intersection of a plane and a double-napped cone. When the plane does pass through the vertex, the resulting figure is a degenerate conic. Conics.

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Conic Sections

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  1. Conic Sections Circle

  2. Conics • Four Basic Conic Shapes: • Circle • Parabola • Ellipse • Hyperbola • A conic section (or simply conic) is the intersection of a plane and a double-napped cone. • When the plane does pass through the vertex, the resulting figure is a degenerate conic.

  3. Conics • Geometrically, each conic is defined as a locus (collection) of points satisfying a geometric property. • Algebraically, each conic is given by a general second-degree equation: • Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 • But in our class, most of time, B = 0, so: • Ax2 + Cy2 + Dx + Ey + F = 0

  4. Circle • A circle is the set of all points in a plane that are a fixed distance r (called the radius) from a fixed point (called the center). • Vocabularies: • Center • Radius • Diameter • Standard Form of a Circle: • (x – h)2 + (y – k)2 = r2 • Center: (h, k) • Radius = r

  5. Practice Problem • Find the equation of the circle with center (–2, 1) and radius 3. • (x+2)2 + (y–1)2 = 9 (Standard Form) • x2 + y2 + 4x – 2y – 4 = 0 (General Form)

  6. Practice Problem • Find the equation of the circle with center (3, 5) and tangent to the y-axis.

  7. Practice Problem • Write the equation of a circle if the endpoints of a diameter are at (5, 4) and (–1, –2).

  8. Carb: x2 + y2 = infinity2 Fin…

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