Conic Sections

The curves produced by cutting a double cone with a plane are called conic sections. These curves are the boundaries of the sections exposed by the cut.

  • When the plane making the cut is perpendicular o the axis, the curve produced is a circle.
  • When the cutting plane is inclined at an angle between the horizontal and that of the generator, the boundary curve is an ellipse.
  • When the cutting plane is parallel to a generator the resulting curve is a parabola.
  • If the cutting plane meets both parts of the cone at an angle the curve produced is in two branches and is a hyperbola.

The conics of various kinds have a common property : There is a fixed point called the focus and a fixed line called the directrix, such that for any point on the conic, the distance to the focus and the distance to the directrix have a constant ratio called the eccentricity, i.e. FP/PN = e

The circle can be regarded as the limiting case of an ellipse as the directrix moves further from the origin, e approaches 0, and the focii approach the centre of the ellipse.

Standard Forms of Conics
Conic Circle Parabola Ellipse Hyperbola
Eccentricity e = 0 e = 1 0 < e < 1 e > 1
Standard Equation x2 + y2 = a2 y2 = 4ax (x2/a2) + (y2/b2) = 1 (x2/a2) – (y2/b2) = 1
Tangent to x1 + y1y = a2 y1y = 2a(x + x1) (x1x/a2) + (y1y/b2) = 1 (x1x/a2) – (y1y/b2) = 1

The parametric equations of a circle are, x = a cos θ and y = a sin θ. As θ takes the values in the range 0 ≤ θ < 360° then a point moves around the circle in an anticlockwise direction. A variable such as θ which is used to determine the position of a point on a curve is known as a parameter. In many applications to physics the natural parameter to choose would be time, so that the position of a point is determined by the value of the time.

The parametric equations for a parabola are, x = at2 and y = 2at. The equation of the normal to the parabola is, y + tx = 2at + at3. The polar form of a parabola is L/r = 1 + cos θ (for -π < θ < π)

The parametric equations for an ellipse are, x = a cos θ and y = b sin θ, and an alternate (and often more useful) form of the parametric equations are, (cos θ1/a)x + (sin θ1/b)y = 1

There is one feature of the hyperbola that is not present in any of the other conics. The branches of the hyperbola are ‘hemmed in’ by the lines, y = ± (b/a)x. The hyperbola approaches one of these lines ever more closely but does not touch or cut it. These lines are called the asymptotes. The parametric equations for a hyperbola are, cosh x = (ex + e-x)/2 and sinh x = (ex + e-x)/2

The general equation,

Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0

Includes all the different types of conics. When,

  • H = 0, A = B ≠ 0 → Circle
  • H2 = AB → Parabola
  • H2 < AB → Ellipse
  • H2 > AB → Hyperbola

~ by jamesdow2013 on April 10, 2013.

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