## Hyperbola – Analytic geometry Formulas

### Hyperbola

In analytic geometry a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other. ### Hyperbola with center at the origin:

Hyperbola with center at the origin and transverse axis on the x-axis.

$\frac{{x}^{2}}{{a}^{2}}–\frac{{y}^{2}}{{b}^{2}}=1$

Hyperbola with center at the origin and transverse axis on the y-axis

$\frac{{y}^{2}}{{a}^{2}}–\frac{{x}^{2}}{{b}^{2}}=1$

### Example:

If the latus rectum of a hyperbola is 8 and eccentricity be 3 / √5, then what is the equation of the hyperbola?

2b2 / a = 8 and 3/√5 = √(1 + b2) / a2

4 / 5 = b2 / a2

a = 5, b = 2√5

Hence, the required equation of the hyperbola is (x2 / 25) − (y2 / 20) = 1

4x2 − 5y2 = 100

### Hyperbola with center at any point (h, k):

Hyperbola with center at (h, k) and transverse axis parallel to the x-axis.

$\frac{\left(x–h{\right)}^{2}}{{a}^{2}}–\frac{\left(y–k{\right)}^{2}}{{b}^{2}}=1$

Hyperbola with center at (h, k) and transverse axis parallel to the y-axis.

$\frac{\left(y–k{\right)}^{2}}{{a}^{2}}–\frac{\left(x–h{\right)}^{2}}{{b}^{2}}=1$

### Elements of Hyperbola:

• Center (h, k). At the origin, (h, k) is (0, 0).
• Transverse axis = 2a and conjugate axis = 2b
• Location of foci c, relative to the center of hyperbola.
$c=\sqrt{{a}^{2}+{b}^{2}}$
• Latus rectum, LR
$2\frac{{b}^{2}}{a}$
• Eccentricity, e
$e=\frac{c}{a}>1.0$
• Location of directrix d relative to the center of hyperbola.
• Equation of asymptotes.
$y–k=±m\left(x–h\right)$

where,

m is (+) for upward asymptote and m is (-) for downward.

m = b/a if the transverse axis is horizontal

m = a/b if the transverse axis is vertical

### Example:

Find the transverse axis, the center, the foci and the vertices of the hyperbola whose equation is

16 y2 – x2 =16

Divide all terms of the given equation by 16 which becomes y2 – x2 / 16 = 1

Transverse axis: y axis or x = 0

center at (0 , 0)

vertices at (0 , 1) and (0 , -1)

c2 = 1 + 16 = 17. Foci are at (0 , √17) and (0 , -√17).