What is the Alignment Equation? Alignment equation formula

For the elliptic equation (taking the focus on the x-axis as an example) x 2 a 2+y 2 b 2 = 1 (a>b>0 a is the semi-major axis.

b is the semi-minor axis c is half of the focal length) (it can also be defined as: when the ratio of the distance from the moving point p to the fixed point o and the fixed line x=xo is always less than 1, the straight line is the alignment of the ellipse. ）

Definition of the alignment, the equation of the alignment.

x=a2c (positive semi-axis of x) x=-a2c(negative semi-axis of x).

Let the hyperbolic equation of the coordinates of point p (x0, y0) 0 on the ellipse.

Take the focus on the x-axis as an example) ( x 2 a 2-y 2 b 2=1 (a,b>0) can also be defined as: when the ratio of the distance from the moving point p to the fixed point o and to the fixed line x=xo is ever,000 is 1, the straight line is the alignment of the hyperbola.

The alignment equation x=a2 c x=-a2 c

Let the coordinates of point P on the hyperbola (x0,y0)c a=(xo+p 2) 丨pf丨》1

Parabola. Take the opening to the right as an example) y 2=2px(p>0) (can also be defined as: when the ratio of the distance from the moving point p to the fixed point o and to the fixed line x=xo is always equal to 1, the straight line is the alignment of the parabola. ）

The alignment equation x=-p 2

Let the coordinates of point P on the parabola be (x0,y0)c a=(xo+p 2) 丨pf丨=1

ps: x 2 = 2py (p>0). The alignment equation is y=-p 2).

The nature of the alignment is a conic curve.

The ratio of the distance from any point to a focal point and its corresponding alignment (the focal point and the alignment on the same side of the y-axis) is the eccentricity.

The derivation method of the alignment.

Let the elliptic equation be x a + y b =1 and the focus be f1(c,0),f2(-c,0)(c>0).

Let a(x,y) be a point on the ellipse.

then af1= [(x-c) +y].

Set the alignment to x=f

Then the distance l from a to the alignment is f-x

Let af1 l=e.

x-c)²+y²=e²（f-x)²

Simplification yields (1-e) x -2xc+c +y -e f +2e fx=0

Let 2c = 2e f

If the point is the right vertex, then (c e -a)e=a-c

When e=c a, the above equation holds.

Therefore f=a c

Then the equation is (1-e)x +y =e f -c

Compare with the original elliptic equation.

a²=(e²f²-c²)/(1-e²),b²=e²f²-c²

a²=(c²/e²-c²)/(1-e²),b²=c²/e²-c²

a²-b²=(c²/e²-c²)e²/(1-e²)=c²

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in conic curves.

, the ratio of the distance from a point to a fixed point to the distance from this point to a fixed line is a fixed value.

The fixed point is the focal point ((c,0) or (0,c)), and the fixed line is the alignment, the alignment equation.

It is used to represent the alignment (x = a c or y = a c).

The ratio is the eccentricity (denoted by e, 0 e 1 trajectory is an ellipse, e 1 trajectory is hyperbola.

Alignment equation x=a2 c (positive half axis of x) x=-a 2 c (negative half axis of x).

Let the coordinates of point p on the ellipse (x0, y0)00) also be defined as: when the ratio of the distance from the moving point p to the fixed point o and to the fixed line x=xo is everten to 1, the straight line is the quasi-line of the hyperbola. ）

The alignment equation x=a2 c x=-a2 c

Let the coordinates of point P on the hyperbola (x0,y0)c a=(xo+p 2) 丨pf丨》1

Parabola (taking the opening to the right as an example) y 2=2px(p>0) (can also be defined as: when the ratio of the distance from the moving point p to the fixed point o and to the fixed line x=xo is always equal to 1, the straight line is the alignment of the parabola. ）

The alignment equation x=-p 2

Let the coordinates of point P on the parabola be (x0,y0)c a=(xo+p 2) 丨pf丨=1

ps: x 2 = 2py (p>0). The alignment equation is y=-p 2).

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