Conic Sections: Circle and Ellipse

• Conics are curves formed by the intersection of a plane and a double cone
• They are a class of geometric shapes that include circles, ellipses, parabolas, and hyperbolas

Circle

• A circle is the set of all points in a plane that are equidistant from a fixed point called the center
• The distance from the center to any point on the circle is called the radius
• Standard form equation: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius
• General form equation: x² + y² + Ax + By + C = 0
• Can be derived from the intersection of a cone with a plane perpendicular to the cone's axis

Ellipse

• An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (foci) is constant
• Each focus is a fixed point
• The major axis is the longest diameter, passing through both foci and the center
• The minor axis is the shortest diameter, perpendicular to the major axis and passing through the center
• Standard form equation: (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis
• a > b for a horizontal ellipse, and a < b for a vertical ellipse
• Foci are located along the major axis, a distance c from the center, where c² = |a² - b²|

Parabola

• A parabola is the set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix)
• The vertex is the point on the parabola closest to both the focus and the directrix
• Axis of symmetry is the line passing through the focus and perpendicular to the directrix
• Standard form equation:
  • Vertical axis: (x - h)² = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus and from the vertex to the directrix
  • Horizontal axis: (y - k)² = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus and from the vertex to the directrix
• The focus is located at (h, k + p) for a vertical axis and at (h + p, k) for a horizontal axis
• The directrix is the line y = k - p for a vertical axis and x = h - p for a horizontal axis

Hyperbola

• A hyperbola is the set of all points in a plane such that the absolute difference of the distances from two fixed points (foci) is constant
• Center is the midpoint between the two foci
• Transverse axis connects the vertices and foci; its length is 2a
• Conjugate axis is perpendicular to the transverse axis and passes through the center; its length is 2b
• Vertices are the points where the hyperbola intersects the transverse axis
• Asymptotes are lines that the hyperbola approaches as it extends to infinity
• Standard form equation:
  • Horizontal transverse axis: (x - h)²/a² - (y - k)²/b² = 1, where (h, k) is the center
  • Vertical transverse axis: (y - k)²/a² - (x - h)²/b² = 1, where (h, k) is the center
• Foci are located along the transverse axis, a distance c from the center, where c² = a² + b²
• Asymptotes' equations: y - k = ±(b/a)(x - h) for a horizontal transverse axis and y - k = ±(a/b)(x - h) for a vertical transverse axis

Eccentricity

• Eccentricity (e) is a parameter that determines the shape of a conic section
• Circle: e = 0
• Ellipse: 0 < e < 1
• Parabola: e = 1
• Hyperbola: e > 1
• For ellipses and hyperbolas, eccentricity is calculated as e = c/a, where c is the distance from the center to a focus and a is the distance from the center to a vertex

General Equation of a Conic Section

• The general form of a conic section equation is Ax² + Bxy + Cy² + Dx + Ey + F = 0
• The discriminant (Δ = B² - 4AC) can determine the type of conic section
  • Δ < 0: Ellipse (or circle if A = C and B = 0)
  • Δ = 0: Parabola
  • Δ > 0: Hyperbola
• If B ≠ 0, the conic section is rotated

Applications of Conics

• Circles: Wheels, gears, and circular motion
• Ellipses: Planetary orbits (Kepler's laws), whispering galleries
• Parabolas: Reflectors in telescopes and spotlights, projectile motion
• Hyperbolas: Navigation systems (LORAN), cooling towers in nuclear power plants

Explanation of conic sections, focusing on circles and ellipses. Includes definitions, key characteristics such as center, radius, foci, major and minor axes. Also includs standard form equations.