The derivations of standard equations in rectangular coordinates.

1. Let's derive the equation for an ellipse. The defining property of an ellipse is that the sum of distances from any point on the curve to two fixed points, called foci is a constant. Let's choose the system of coordinates so that foci will have coordinates (-c,0) and (c,0). Let their name be F1 and F2. Then we have for any point P(x,y) on the curve PF1 + PF2 = 2a for some a>c. It follows that
(x+c)2 +y2 + ((x-c)2 +y2) = 2a

Let's separate the radicals and square the equation:

x2 + 2xc +c2 + y2 = a2 - 4a ((x-c)2 + y2) +x2 -2xc +c2 + y2

After simplifying, separating the radical and dividing by 4 we get:

a ((x-c)2 +y2) = a2 - xc

We then square the equation again and simplify:

a2(x2 -2xc +c2 + y2) = a4 - 2a2xc + x2c2
x2(a2-c2) + a2y2 = a4 -a2c2

Let's remember that a>c and b2 = a2 + c2. Then c2 = a2 - b2. Therefore

x2b2 +y2a2 = a2b2

Or,
x 2 + y 2 = 1
a2 b2


2. For the derivation of a hyperbola, virtually the same derivation holds true. The only difference is that PF1 - PF2 = 2a for some a<c. Since a<c, c2 = a2 + b2. With the exception of these differences the derivations are basically the same.

Note: If P is not on the origin let Px = h and Py= k and then substitute for x; x-h and y x-k).




3.Now let's derive the equation for a parabola. The defining property of a parabola is that the sum of distances from any point on the curve to the fixed point, called the focus, and to the fixed line, called the directrix is constant. Let's choose the system of coordinates so that the focus will have the coordinate (p/2,0) and the equation of the directrix will be x= -p/2. There names will be F and D respectively. Then we have for any point P(x,y) on the curve PF + PD =p.

It follows that

x + p = ((x-p 2)2 +y2)
2 2

We then square this equation:

x2 + px + p 2 = x2 -px +p 2 + y2
4 4

Or,

y2 +2px



4. Now let's derive the equation for a circle. The defining property of a circle is that the distance from any point P(x,y) on the circle to the center (h, k) is a constant r.

Thus,

((x-h)2 + (y-k)2) = r

Square both sides

(x-h)2 + (y-k)2= r2