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¶
The equation of a circle centered at the origin with radius \(r\) is given by
\[x^2+y^2=r^2\]The equation of a circle centered at the point \((h,k)\) with radius \(r\) is given by
\[(x-h)^2+(y-k)^2=r^2\]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.
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
\[y - 1 = \frac{1}{4}x^2\]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
\[y - k = \frac{1}{4k}x^2\]