The equation of pair of asymptotes differ from equation of hyperbola and its conjugate by

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Solution

Consider general equation of hyperbola,
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$

The equation of pair of asymptotes can be written as let's say $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c^{'} = 0 \dots (1)$

For any $2^{n d}$ degree curve $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ to represent pair of straight lines, we have: $△ = a b c + 2 f g h - a f^{2} - b g^{2} - c f^{2} = 0$

Equation $(1)$ represents pair of straight lines.
$△ = a b c^{'} + 2 f g h - a f^{2} - b g^{2} - c^{'} f^{2} = 0$

So, we get value of $c^{'}$ and hence the equation of pair of asymptotes of this general hyperbola.

We can see that on comparing the hyperbola equation with the pair of straight lines equation, the only difference is between the constants.

Hence, the equation of pair of asymptotes differ from equation of hyperbola and its conjugate by constant.

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