The conic sections of a circle, parabola, ellipse and hyperbola

The conic sections of a circle, parabola, ellipse and hyperbola

Conic Sections

Conic sections are the curves obtained by intersecting a right circular cone with a plane. They include four types: circles, ellipses, parabolas, and hyperbolas.

Circle

  • A circle is a set of points in a plane that are a fixed distance from a fixed point.
  • The fixed point is called the centre of the circle.
  • The fixed distance is called the radius of the circle.
  • The standard form of a circle’s equation is (x-h)² + (y-k)² = r², where (h,k) are the coordinates of the center and r is the radius.

Ellipse

  • An ellipse is a set of points where the sum of the distances from two fixed points, called foci, is constant.
  • The major axis is the largest diameter that cuts through the centre of the ellipse.
  • The minor axis is the smallest diameter that cuts through the centre of the ellipse.
  • The standard equation of an ellipse centered at the origin is x²/a² + y²/b² = 1, where a is the semi-major axis length and b is the semi-minor axis length.

Parabola

  • A parabola is a set of points in a plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix.
  • The line passing through the focus and perpendicular to the directrix is called the axis of symmetry.
  • The standard equation of a parabola is y² = 4ax, where a is the distance from the origin to the focus.

Hyperbola

  • A hyperbola is a set of points where the difference of the distances from two fixed points, called foci, is constant.
  • A hyperbola has two disconnected curves called branches.
  • The transverse axis is the line segment joining the two vertices of the hyperbola.
  • The standard equation of a hyperbola centered at the origin is x²/a² - y²/b² = 1 or y²/b² - x²/a² = 1, depending on the orientation, where a and b are related to the distance from the centre to the vertices and the center to a point on the asymptote, respectively.

Conic sections are an essential concept in Further Pure Mathematics, so it’s critical to understand their defining properties, equations, and how they relate to each other.