BodhiAI

Nov. 13, 2019

**Intersection of Conjugate Diameters and the Hyperbola**

**To prove that of a pair conjugate diameters of a hyperbola, only one meets the curve in real points. **

**Let y = mx .......(1)**

**and y = m1x .......(2)**

**be a pair of conjugate diameters of the hyperbola then **

** .......(3)**

**Solving (1) and the equation of hyperbola we get **

** **

**or ......(4)**

**Similarly (2) meets the hyperbola at points whose abscissa are given by **

** .......(5)**

**The two values of x given by (4) will be real **

**if i.e. **

**i.e., **

** i.e., if **

**i.e., if **

**Then form (5), the values of x are imaginary.**

**Hence if (1) meets the hyperbola in real points then (2) meets it in imaginary points and vice-versa. **

**Now, if CD is the conjugate diameter of a diameter CP of the hyperbola where P is then equation of CP, where C is (0, 0), is**

** **

** **

**Comparing this with y = mx, we have **

** **

** **

** **

**Hence, equation of conjugate diameter is **

** **

**Solving the equations cosec and the coordinates of D are **