## Solving Lesson 3 Page 56 Math Learning Topic 10 – Kite>

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