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Elliptic Curve – Hyperbola
Motivation:
For ellipse, we know that it is the set of points that the sum of the distances to two fixed points is a constant. How about the set of points that the difference of the distances to two fixed points is a constant? We start looking at it by finding its equation first.
Definition ( Hyperbola ) Given two fixed points and a constant c>0, hyperbola is the set of all the points that the difference of the distance to the two points is equal to the constant c.
Similarly, we call the two fixed points “foci” ( plural of focus ).
Let’s start with the following setting: Two fixed points: F1( d,0 ) , F2(-d,0) , c>0, d>0, c < 2d
| - | = c (x-d)2+ y2 - 2 + (x+d)2+y2 = c2 2x2+2d2+2y2 – c2 = 2 (2(x2+y2)+ (2d2-c2))2 = 4 ( (x-d)2+y2) ( (x+d)2 + y2)
4(x2+y2)2 + 4(x2+y2)(2d2-c2) + (2d2-c2)2 = 4((x2-d2)2 + 2(x2+d2)y2 + y4)
4(x4+2x2y2+y4)+4(x2+y2)(2d2-c2) + (2d2-c2)2 = 4(x4-2x2d2+d4+ 2(x2+d2)y2 + y4)
4(x2+y2)(2d2-c2) + (2d2-c2)2 = -8x2d2 + 4d4 + 8d2y2
x2( 16d2-4c2) –4c2 y2=4d2c2-c4
In other words, if we get an equation like
then focus: ( ,0) , (- ,0) The curve does not cross Y-axis. The curve crosses X-axis at (a,0) and (-a,0) .
Similarly, if we get an equation like
then focus: (0, ) , (0,- ) The curve does not cross X-axis . The curve crosses Y-axis at (0,b) and (0,-b) .
Theorem ( Tangent Line ) If a point P(c,d) is on the hyperbola , a, b R then the slope m of the tangent line satisfies
Proof: Similar approach being used for ellipse can be also used here. All the expression are almost the same other than replacing b2 with –b2 . Please refer to the previous section.
Asymptotic Line of Hyperbola ( Asymptotes ) The concept of asymptotic line to approximate the behavior of a function or curve . The hyperbola with the equation
or
has two asymptotic lines bx – ay=0 and bx+ay=0 .
Sol:
It can be considered in the following way:
while y 0 while x and y are very large, the term is very small such that
Thus, we have asymptotic lines bx – ay=0 and bx+ay=0 .
Similarly, the same approach can be used to find the asymptotic lines of the hyperbola with another equation.
Theorem ( Reflection ) As shown below, F1 and F2 are the foci of the hyperbola , a>0, b>0 The point P(c,d) is on the hyperbola. And the line L is the tangent line of the hyperbola such that the line crosses the hyperbola at point P. Then
1 = 2
The physical meaning of this property is: if you construct a mirror along the outer rim of the hyperbola , when we shoot the light toward the focus F2 from outside, the light will be reflected toward F1 by the hyperbola mirror. In the long run, the light will be bounced back and forth along the axis of the hyperbola if the energy of the light is not lost anywhere.
Proof: The proof is similar to what we have done for ellipse. Find the normal vector of the tangent line first. And then check and (This is the way for the angle with respect to normal vector. )
We start with listing all the basic elements of the hyperbola:
Hyperbola:
Foci : F2( ,0) , F1(- ,0) tangent line L : y = (x-c) + d Normal vector of line L : =(b2c, -a2d)
=(c- , d) =(- -c, -d) = = =
Point P is on the hyperbola. Thus, d2 = b2( -1) . = =
Please note that
= = = | |
Thus, = =
= ((c- ) +d ) =( 1- ) =
=((- -c) -d ) =(-1- ) =
= It means their angles with respect to normal vector of the tangent line are equal. 1 = 2
From the derivation here, we even can know the relationship between the angle and its position. |
