Monday, April 28, 2008

Conic Sections - Hyperbola

Hyperbola
Equation x²/a² - y²/b² = 1

Some important results connected with x²/a² - y²/b² = 1

1. Vertex A(a,0) and A’(-a,0), AA’ is called the transverse axis whose length is 2a.
2. Focus S
3. Foot of the directrix Z
4. Equation of directrix
5. Equation of latus rectum (L.R.) x = ae
6. Equation of axis
7. Equation of tangent at vertex
8. Length of latus rectum = 2b²/a
9. Extremities of L.R = (ae,b²/a) and (ae, -b²/a)
10. Focal distance of any point (x,y)

Most of the equations for various elements like tangents, normals etc. of the ellipse x²/a² + y²/b² = 1 are true for the hyperbola x²/a² - y²/b² = 1 if we replace b² by -b².

1. Tangent at (x1,y1) is

xx1/a²-yy1/b² = 1

2. Normal at (x1,y1) is

x-x1/(x1/a²) = -(y-y1)/(y1/b²)

3. The line y = mx+c is a tangent to the hyperbola if c² = a²m²-b² .
4. Hence, the line y = mx+SQRT(a²m²-b²) is always a tangent.
5. The points of contact are (-a²m/c,-b²/c)
6. The lx + my = n will be a tangent to the hyperbola is

a²l² = b²m² = n²
7. From any pont (h,k) outside the hyperbola two tangents (real) can be drawn . The slopes are given by the equation
m²(h²-a²) – 2mhk + (k²+b²) = 0
8. Polar of any point (h,k) in relation to the hyperbola is

hx/a²-ky/b² = 1