BodhiAI

Nov. 12, 2019

**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 x ^{n}is present in the equation of the curve, then there is no Asymptote parallel to x-axis.**

**Step III If the term of x ^{n} is missing, then equate to zero the coefficient of x^{n-1}this 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, x ^{3} 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 **

x^{2}/a^{2} - y^{2}/b^{2} = 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 **

** **