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
Solving (1) and the equation of hyperbola we get
Similarly (2) meets the hyperbola at points whose abscissa are given by
The two values of x given by (4) will be real
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