First slide

Hyperbola in conic sections

Question

If a diameter of a hyperbola meets the hyperbola in real points then

Moderate

A

it meets the conjugate hyperbola in imaginary points.
B

the conjugate diameter meets the given hyperbola in real point
C

he conjugate diameter meets the conjugate hyperbola in imaginary points.
D

none of these

Solution

Equation of the hyperbola be $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ so equation of the conjugate hyperbola is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = - 1$
Equation of a diameter of the hyperbola is $y = (b^{2} / a^{2} m) x$ which meets the hyperbola at points given by
$\frac{x^{2}}{a^{2}} - \frac{b^{4} x^{2}}{b^{2} a^{4} m^{2}} = 1 \Rightarrow x^{2} = \frac{a^{2} m^{2}}{m^{2} - b^{2}}$
These points are real if $m^{2} > b^{2}$ and for this value of m, the diameter will meet the conjugate hyperbola at point for which $x^{2} = \frac{- a^{2} m^{2}}{m^{2} - b^{2}}$ and these points are imaginary.
Note: Conjugate diameter will meet the conjugate hyperbola in real points and the hyperbola in imaginary points

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