BodhiAI
Nov. 13, 2019

#### Auxiliary Circle, Director Circle& Parametric Equation of standard Hyperbola:

Director Circle

The locus of the point of intersection of the tangents to the hyperbola which are perpendicular to each other is called the Director Circle.

If the equation of hyperbola is

then the equation of director circle to it is

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 xn is 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-1 this will give the asymptote parallelto x-axis.

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

Example: a) Let the equation of the curve be

Since, the degree of equation is 3.

Now, 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),  (x2/a2) - ((mx+c)2/b2) = 1      ......(2)

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.