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BodhiAI
Nov. 12, 2019

Rectangle Hyperbola ,Assymptotes:

 

Asymptotes

Asymptotes are the lines which are tangents to the curve at infinity. In other words the branch of the curve moves almost parallel to Asymptote.

To find the equation of Asymptotes of any equation of the curve, we can use the following steps.

Step I Determine the degree of the equation of the curve. Let is is n.

Step II If xnis present in the equation of the curve, then there is no Asymptote parallel to x-axis.

Step III If the term of  xn is missing, then equate to zero the coefficient of xn-1this will give the asymptote parallelto x-axis.

Similarly, we can find the saymptote parallel to y-axis.

Example: a) Let the equation of the curve be 

Since, the degree of equation is 3. 

Now, x3 is present in the equation, So there is  no asymptote along x-axis. Also y3 is not present in the equation byt y2 is present in the equation, so there is one asymptote parallel to y-axis and i.e. x+a = 0.

b) Let the equation of the curve be

Since, the degree of the equation is 4 and both x4 and x3 are missing. Hence, equate to zero the coefficient of y2 i.e. x2 -a2 = 0

Hence x = +-a are the Asymptotes parallel to y-axis.

Equation of Asymptotes of Hyperbola

Let y = mx+c be an asymptote of the hyperbola 

x2/a2 - y2/b2 = 1     .........(1)

Substituting the value of y in (1),

If the line y = mx+c is an asymptote to the given hyperbola, then it touches the hyperbola at infinity. So both roots of equation (2) must be infinite. 

and

then and c = 0

Substituting the value of m and c in y = mx+c, we get

Hence, the two degree equation in two variables 

represents the equation of pair of Asymptotes, which is a two degree equation in two variables x and y passing through origin.

Properties of Asymptotes of Hyperbola

P-1 A hyperbola and its conjugate hyperbola have the same asymptotes.

P-2 The angle between the asymptotes of is 2

P-3 The asymptotes pass through the centre of the hyperbola.

P-4 The bisectors of the angles between the asymptotes are the co-ordinates axes. 

P-5 Let

and

be the equations of the hyperbola, asymptotes and the conjugate hyperbola respectively, then clearly C + H = 2A

P-6 The asymptotes of rectangular hyperbola which are at right

Rectangular Hyperbola

A hyperbola whose asymptotes include a right angle is said to be rectangular hyperbola. Also, if the lengths of transverse and conjugate axes of any hyperbola be equal, it is called rectangular or equilateral hyperbola.

Equation of Rectangular Hyperbola 

Since, the Asymptotes of the rectangular hyperbola are at right angles, then 

{Angle between asymptotes}

⇒b/a=1

∴ a=b

Thus, the equation of rectangular hyperbola is