**Let F(c,0) be the focus**, with **c > 0 and x = c / e²** be the directrix of the hyperbola, also Let **P(x, y)** be a point on the hyperbola, by definition **|PF| / |PM| =e ** is called** standard equation of hyperbola** in math proof ↓

that is,

**(x – c)² + y ² = e² (x – c / e ²)**

**x ² – 2 x c + c ² + y ² = e ² x ² – 2 c x + c² / e ²**

**x ² (e ² -1) – y ² = c ² (1 – 1 / e²)**

**c ² / e ² (e ² -1) ……………… B**

Let us set a = c / e so that equation B become

**x ² (e ² -1) -y ² – a ² (e ² -1) = 0**

**x ² / a ² – y ² / a ² (e ² -1) =1**

**x ² / a ² – y ² / b ² =1 …………….. (C)**

**where b ² = a ² (e ² -1) = c ² – a ² because c = a. e**

**Equation (C) is a standard form of the hyperbola.**

It is clear that the curve is symmetric with respect to both the axis.

**If we take the point (-c, 0) as focus and the line x = -c / e ² as directrix**

Then it is easy to see that the set of all point P(x, y) such that

**|PF| = e |PM| is hyperbola Form equation=x ² / a ² – y ² / b ² =1**

**thus, a hyperbola has two foci and two direct rices.**

**If the Foci lies on y-axis, then role of x and y are interchange in x ² / a ² – y ² / b ² =1**

**then the equation of parabola become**

standard equation of hyperbola math

y ² / a ² – x ² / b ² =1

**Definition:**

The hyperbola

**x ² / a ² – y ² / b ² =1**

meets the x-axis at a point with y = 0 and x = ± a. the point A(-a, 0) and A'(a, 0)

are called the **vertices of the hyperbola.**

The line segment AA’ = 2Aa is called the** Focal axis (transverse)** of hyperbola, x ² / a ² – y ² / b ² =1

The hyperbola x ² / a ² – y ² / b ²=1 does not meet the** y-axis in real point****.**

However, the line segment joining the point B(0, -b) and B'(0, b) is called the

**Conjugate axis of the hyperbola**.

**The mid-point (0, 0) of AA’ is called the center of the **

**hyperbola**

In case of Hyperbola,

** x ² / a ² – y ² / b ² =1 **

we have b ² = a ² (e ² -1)= c² – a²

**The eccentricity e = c / a > 1**

So that

Unlike the ellipse

**We may have b > a or b < a or b = a**

**Parametric Equation of hyperbola:**

**the point (a sec θ, b Tan θ) lies on the hyperbola x ² / a ² – y ² / b ² =1**

for all real value of θ. The Equation **x = a sec θ, y = b tan θ ** are called parametric

**Equation of Hyperbola.**

**Asymptotes of a curve:**

**Since y = ± b / a √ x² – a ² modulus of |x| is very large**

**so that x ² – a² → x ² we have**

**y = ± b / a (x) that is x ² / a² – y ² / b ² …………. L**

These lines L do not meet the curve but distance of any point on the curve from

any of the two lines approaches to zero. Such lines are called **ASYMPTOTES** of a curve.

**Central Conic**:

The ellipse and hyperbola are called central conic because each has a center of symmetry>

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