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BodhiAI
Nov. 13, 2019

Intersection of Hyperbola With Circle /Ellipse/ParabolaIntersection of Hyperbola With Circle /Ellipse/Parabola:

 

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