How to calculate the focal length of hyperbola?

What are (c, 0) and (-c, 0) on the X-axis?

On the y axis are (0, c) and (0, -c) respectively?

C= radical sign (a 2+b 2)

We call the locus whose absolute value is equal to a constant (constant is 2a, less than |F 1F2|) a hyperbola. The locus of a point whose absolute value of the distance difference between two points on a plane is a fixed length is called hyperbola.

Namely: │ | pf1| | pf2 │ | = 2a.

Definition 1:

On the plane, the locus of a point whose absolute value is constant (less than the distance between two fixed points) is called hyperbola. This fixed point is called the focus of hyperbola.

Definition 2: The ratio of the distance to a given point to the distance of a straight line on the plane is a constant e ((e >; 1), that is, the eccentricity of hyperbola) is called hyperbola. The fixed point is called the focus of hyperbola, and the fixed line is called the directrix of hyperbola. The equation of hyperbola directrix is (focus on X axis) or (focus on Y axis).

Definition 3: Plane cutting conical surface. When the cross section is not parallel to the generatrix of the conical surface and does not pass through the vertex of the conical surface, and the two conical surfaces of the conical surface intersect, the intersection line is called hyperbola.

Definition 4: In the plane rectangular coordinate system, when the binary quadratic equation F(x, y)=ax2+bxy+cy2+dx+ey+f=0 meets the following conditions, it is hyperbolic.

1, a, b and c are not all zeros.

2、δ= B2-4ac >0。

Note: Article 1 can be inferred from Article 2.

Analytic geometry in high school, I learned that the center of hyperbola is at the origin, like symmetry about X and Y.

The above four definitions are equivalent, and it is judged that the image is symmetrical about X and Y according to the position before and after construction.

The standard formula is:

1, when the focus is on the x axis:? (a & gt0，b & gt0)

2. When the focus is on the Y axis, it is:? (a & gt0，b & gt0)

Extended data:

Value range

│x│≥a (focusing on the X axis) or │y│≥a (focusing on the Y axis).

symmetrical

Symmetry about the coordinate axis and the origin, where the center is symmetrical about the origin.

pinnacle

A(-a, 0), A'(a, 0). At the same time, AA' is called the real axis of hyperbola, and │AA'│=2a.

B(0, -b), B'(0, b). At the same time BB' is called the imaginary axis of hyperbola, and │BB'│=2b.

F 1(-c, 0) or (0, -c), F2(c, 0) or (0, c). F 1 is the left focus of hyperbola, F2 is the right focus of hyperbola and │F 1F2│=2c.

For real axis, imaginary axis and focus, a2+b2=c2.

Asymptote

The focus is on the x axis:.

Focus on the y axis:

Conic curve ρ=ε/ 1-ecosθ when E >: 1 represents hyperbola. Where p is that distance from the focus to the directrix and θ is the angle between the chord and the X axis.

Let 1-ecosθ=0 to find θ, which is the inclination of the asymptote, that is, θ=arccos( 1/e).

Let θ=0, and we get ρ=ε/( 1-e) and x=ρcosθ=ε/( 1-e).

Let θ = π, and we get ρ=ε/( 1+e) and x=ρcosθ=-ε/( 1+e).

These two x's are the abscissa of the fixed point of hyperbola.

Find the abscissa of their midpoint (the abscissa of the hyperbola center)

x =[(ε/ 1-e)+(-ε/ 1+e)]/2

(Pay attention to simplification)

The straight line ρ cos θ = [(ε/1-e)+(-ε/1+e)]/2

It is the symmetry axis of hyperbola. Note that it is an axis of symmetry that does not intersect the curve.

Rotate the line clockwise by π/2-arccos( 1/e) to get the asymptote equation, and let the rotation angle be θ'.

Then θ′ = θ-[π/2-arccos (1/e)]

Then θ=θ'+[π/2-arccos( 1/e)]