Equation of Ellipses and Circles
Summary:
This lesson explains the derivation of the equation of ellipses from their geometric definition, which states that the sum of the distances from any point on the ellipse to two fixed foci is constant. Through a detailed algebraic development, the general equation of ellipses and their canonical form is deduced, as well as the connection between ellipses and circles, showing that a circle is a special case of an ellipse when the semi-axes are equal.
Learning Objectives:
At the end of this lesson, the student will be able to:
- Deduce the equation of ellipses from their geometric definition.
- Recognize both the general form and the canonical form of the ellipse equation.
CONTENT INDEX
Geometric Formulation
Derivation of the Ellipse Equation
General Equation of Ellipses
Canonical Equation of Ellipses
Reduction to the Circle Equation
Geometric Formulation
To obtain the equation that describes ellipses, we must reason, as with parabolas, about the geometric meaning of these. An ellipse is the set of all points in the plane such that the sum of the distances between them and two points called foci is always the same.
That is, it will satisfy:
d(f_1,p) + d(f_2,p) = constant
Derivation of the Ellipse Equation
From the geometric definition of ellipses, we can obtain an algebraic expression that describes them. To do this easily, however, we will resort to some simplifications. We will consider, without loss of generality, that the foci are located at f_1 =(-c,0) and f_2 =(c,0), so that if any point p=(x,y) is part of the ellipse, then it must satisfy that:
\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a
Where a\in\mathbb{R} is a fixed constant. From this, we can build the following reasoning:
| (1) | \sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a | ; Geometric definition of ellipse |
| \sqrt{(x-c)^2 + y^2} = 2a - \sqrt{(x+c)^2 + y^2} | ||
| (2) | (x-c)^2 + \cancel{y^2} = 4a^2 -4a \sqrt{(x+c)^2 + y^2} + (x+c)^2 + \cancel{y^2} | ; squaring (1) |
| (x-c)^2 = 4a^2 -4a \sqrt{(x+c)^2 + y^2} + (x+c)^2 | ||
| \cancel{x^2} -2xc + \cancel{c^2} = 4a^2 -4a \sqrt{(x+c)^2 + y^2} + \cancel{x^2} +2xc + \cancel{c^2} | ||
| -2xc = 4a^2 -4a \sqrt{(x+c)^2 + y^2} +2xc | ||
| 4a \sqrt{(x+c)^2 + y^2} = 4a^2 +4xc = 4(a^2 + xc) | ||
| a \sqrt{(x+c)^2 + y^2} = a^2 + xc | ||
| (3) | a^2 [(x+c)^2 + y^2] = (a^2 + xc)^2 | ; squaring (2) |
| a^2 [x^2 + 2xc + c2 + y^2] = a^4 +2a^2xc + x^2c^2 | ||
| a^2 x^2 + \cancel{2xca^2} + a^2 c2 + a^2 y^2 = a^4 + \cancel{2a^2xc} + x^2c^2 | ||
| a^2 x^2 + a^2 c2 + a^2 y^2 = a^4 + x^2c^2 | ||
| x^2 (a^2 - c^2) + a^2 y^2 = a^4 - a^2 c^2 =a^2(a^2-c^2) | ||
| \dfrac{x^2}{a^2} +\dfrac{ y^2}{a^2-c^2} = 1 | ||
| (4) | 0\lt a^2 - c^2 =: b^2 | ; The number represented by b^2 is positive, as seen from the figure. |
| (5) | {\dfrac{x^2}{a^2} +\dfrac{ y^2}{b^2} = 1} | ; From (3) and (4) |
| \boxed{\left(\dfrac{x}{a}\right)^2 + \left(\dfrac{y}{b}\right)^2 = 1} |
This is what we call the “ellipse equation”.
General Equation of Ellipses
The equation we just obtained can be brought to its general form through translation transformations by substituting x\longmapsto (x-h) and y\longmapsto (y-k). This leads us to the general form of the ellipse equation:
\boxed{\left(\dfrac{x-h}{a}\right)^2 + \left(\dfrac{y-k}{b}\right)^2 = 1}
This is an ellipse centered at the point (h,k)
Canonical Equation of Ellipses
By doing algebra on this, we arrive at the canonical equation of ellipses:
| (1) | \left(\dfrac{x-h}{a}\right)^2 + \left(\dfrac{y-k}{b}\right)^2 = 1 | ; General ellipse equation |
| b^2 (x-h)^2 + a^2(y-k)^2 = a^2 b^2 | ; Multiply everything by a^2b^2 | |
| b^2 [x^2-2xh+h^2] + a^2[y^2-2yk + k^2] = a^2 b^2 | ; Expand squares | |
| b^2 x^2-2hb^2 x + h^2b^2 + a^2 y^2-2ka^2y + k^2a^2 = a^2 b^2 | ; Expand parentheses | |
| b^2 x^2- 2hb^2 x + a^2 y^2-2ka^2y +(h^2b^2 + k^2a^2 - a^2 b^2) = 0 | ; Group constant terms |
In this last expression, we can make the substitutions A:=b^2, B:=-2hb^2, C:=a^2, D:=-2ka^2 and E:=h^2b^2 + k^2a^2 - a^2 b^2. Thus, we will see that ellipses will be described by equations of the form:
Ax^2 + Bx + Cy^2 + Dy + E = 0
This is what we call the “Canonical Equation of Ellipses”.
From these developments, some restrictions on the constants of the canonical equation can be extracted. The most important is that A and B must have the same sign; otherwise, we would no longer be talking about an ellipse but rather a hyperbola. There are more restrictions on the constants of the canonical representation, but discussing them now is not the most efficient; we will go over them in detail when we review the characterization of ellipses and hyperbolas.
Reduction to the Circle Equation
One thing we will review when we talk about the characterization of ellipses is that the constants a and b in the general equation correspond to the semi-axes of the ellipse. If we take both semi-axes and equalize them, making a=b=r, then the ellipse will become a circle with radius r.
General Equation of Circles
In this way, the general equation of circles is obtained as:
(x-h)^2 + (y-k)^2 = r^2
Canonical Equation of Circles
Similarly, the canonical equation of circles is obtained
Ax^2 + Bx + Cy^2 + Dy + E = 0
In its canonical form, it coincides with ellipses, as we have seen, since circles, as we have seen, are a particular case of ellipses.