Topic
In the coordinate plane Oxy, for a hyperbola the canonical equation is \({x^2} – {y^2} = 1\). Prove that the two asymptotes of the hyperbola are perpendicular to each other.
Solution method – See details
Equation of hyperbola \(\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1\” ) where \(a > 0,b > 0\). Then we have:
+ The two asymptotes of the hyperbola (H) have the equation \(y = – \frac{b}{a}x,y = \frac{b}{a}x\)
Detailed explanation
We have \(a = 1,b = 1\) so we have the equations of the two asymptotes of the hyperbola (H) with the equation \(y = – x,y = x\ respectively).
These two lines have slopes of \({k_1} = – 1;{k_2} = 1\)
We see \({k_1}.{k_2} = – 1\) so these two lines are perpendicular to each other
