Page 56 - 4263

Lines of the second order

Then (7.13) looks like:

√ √
2 2 2 2
(x + c) + y − (x − c) + y = 2a. (7.15)
Equation (7.15) is the equation of the hyperbola, but in order to make it comfortable to use,
we will perform elementary transformations in an analogy with those, made in subsection 7.2:

y M(x, y)

τ 1 τ 2

O . . . . . . . . .
F 1 (−c, 0) F 2 (c, 0) x

Figure 7.6 – Construction of the hyperbola

√ √ √
2
2
2
2
2
2
2
2
2
2
(x+c) +y = 4a ±4a (x − c) + y + (x − c) + y or cx−a = ±a (x − c) + y .
4
2 2
2 2
2
2 2
2
2
2
2 2
2
2
2
2 2
2
c x − 2a cx + a = a x − 2a cx + a c + a y , (c − a )x − a y = a (c − a ).
2
2
As c > a, then c − a > 0. Let’s write:
√
2
2
2
2
2
b = c − a or b = c − a . (7.16)
Then
x 2 y 2
2 2
2 2
2 2
b x − a y = a b or − = 1. (7.17)
a 2 b 2
In an analogy to the ellipse, the equivalence of equations (7.15) and (7.17) can be proved.
Equation (7.17) is called the canonical equation of the hyperbola.
Let’s study the shape of the hyperbola following 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, the part of the hyperbola, located in the first quarter, will be considered.
For that part y ≥ 0, and so, from equation (7.17):
b √
2
2
y = x − a . (7.18)
a
From this equation the following conclusions can be made:
1. for 0 ≤ x < a the hyperbola doesn’t exist.
2. if x = a then y = 0, so point (a; 0) belongs to the hyperbola. This is point
A on (fig. 7.7).
3. if x > a, then y > 0, moreover, y increases (y → ∞) while x increases
x → ∞, i.e. point M moves to the infinity.
In order to find out this, let’s discuss equation:
b
y = x. (7.19)
a
This is the equation of the straight line, which crosses the beginning of coordinates, with
b
angle coefficient k = . The part of that line, located in the first quarter, is shown on (fig. 7.7).
a
56