Contents

A.4 Conic Sections

We can derive basic formulas for the conic sections using simplifications, then generate more robust formulas later once we have worked the details of transformations.

Circle

The set of all points equidistant from a point called the center.

Radius

The common distance from a point on a circle to its center.

Parabola

A set of points which are equidistant from a focal point \(F\) and a line \(l\) called the directrix.

Hyperbola

A set of points where the absolute difference of the distances from two points (called the foci) is constant.

Theorems

  1. The equation of a circle centered at the origin with radius \(r\) is given by

    \[x^2+y^2=r^2\]
  2. The equation of a circle centered at the point \((h,k)\) with radius \(r\) is given by

    \[(x-h)^2+(y-k)^2=r^2\]
  3. For a given parabola, drop a perpendicular from the focal point \(F\) to the closest point on the directrix \(P\). Prove the midpoint of \(FP\) is on the parabola.

  4. Suppose we have a parabola the direterix of which is the \(x\)-axis with focal point \(F=(0,2)\). Prove that the set of all points \(P=(x,y)\) on the parabola is equivalent to

    \[y - 1 = \frac{1}{4}x^2\]
  5. Suppose we have a parabola the direterix of which is the \(x\)-axis with focal point \(F=(0,k)\). Prove that the set of all points \(P=(x,y)\) on the parabola is equivalent to

    \[y - k = \frac{1}{4k}x^2\]