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Ellipse

Equation (7.6) is the equation of the ellipse, but in this view it is uncomfortable to use. Let’s
√
2
2
2
2
2
simplify by means of elementary transformations: (x + c) + y =4a − 4a (x − c) + y +
√ √ 2
2
2
2
(x − c) + y or a (x − c) + y = a − cx. Square again:
2
2 2
4
2
2 2
2
2 2
2 2
a x − 2a cx + a c + a y = a − 2a cx + c x .
2 2
2
2
2
2
2
2
(a − c )x + a y = a (a − c ). (7.7)
√
2 2
2
2
2 2
2
2 2
Let’s mark: b = a − c or b = a − c . Then formula (7.7) looks like: b x + a y = a b .
2
2
2 2
Having divided both parts of the last equation on a b , we will get:
x 2 y 2
+ = 1. (7.8)
a 2 b 2
As equation (7.8) is obtained from equation (7.6), then coordinates of any point of the ellipse,
that satisfy equation (7.6), will satisfy equation (7.8) as well. When performing basic transfor-
mations, we squared two times. Let’s prove, that equations (7.6) and (7.8) are equivalent. It is
sufficient to prove, that for any point, coordinates of which satisfy equation (7.8), numbers τ 1
and τ 2 satisfy the conditions (7.4) according to the definition of the ellipse. Let’s suppose, that
2 2 2 x 2
coordinates x and y satisfy equation (7.8). Let’s get y from this equation:y = b (1 − a 2 ) and
√
substitute in (7.5). Then τ 1 = (a + cx 2 cx . As |x| < a (which goes from formula
) = a +
a a
(7.8)) and c < 1, then the expression a + cx > 0. That’s why τ 1 = a + cx . In an analogy, τ 2 can
a a a
cx
be calculated: τ 2 = a − . Having added τ 1 and τ 2 , we will get: τ 1 + τ 2 = 2a. Thus, any point,
a
coordinates of which satisfy equation (7.8), belongs to the ellipse and vice versa. Equation (7.8)
is called the canonical equation of the ellipse. This is the second order equation relatively to x
and y.
Let’s study the shape of the ellipse according to its canonical equation. As x and y stand
there in the odd orders, so the curve is symmetric relatively to x−, y− axes and the beginning
of coordinates. Therefore, it is sufficient to get the shape of that part, located in the first quarter.
For that part y ≥ 0, and so, from equation (7.8) :
b √
2
2
y = a − x . (7.9)
a
From equation (7.9) the following conclusions can be made:
1. if x = 0 then y = b, that is point (0; b) belongs to the ellipse. This is point B
on (fig. 7.3).
2. if y = 0 then x = a, that is point (a; 0) belongs to the ellipse. This is point A
on (fig. 7.3).
3. in case x > a, there are no real values for variable y, meaning there isn’t
any point that belongs to the ellipse and satisfies condition x > a.
4. when x increases from 0 to a, then y exists and decreases.
So, the part of that arc of the ellipse, located in the first quarter, is arc BA. Using symmetry of
the ellipse relatively to coordinate axes, we will get the whole ellipse (fig. 7.3).
From the definition of the ellipse a simple method of it’s building can be used. Having
fastened edges of a thread with length 2a in points F 1 and F 2 , strained the thread by the edge
of a pencil, and moving the last, it will build the ellipse with focuses F 1 and F 2 and the sum
of focal radii 2a. So, the ellipse is a convex closed curve. Axes of symmetry of the ellipse are
called its axes, a point of crossing its axes is the center of ellipse.
′
Points of crossing ellipse with coordinates axes are called its tops (points A, A , B, B on
′
(fig. 7.3)).
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