Absolute Asymptotes

Calculus Level 3

How many asymptotes (horizontal and vertical) does the graph of the equation ∣ x y ∣ − ∣ x + y ∣ = 0 \left|xy\right|-\left|x+y\right|=0 have?


Notation: ∣ ⋅ ∣ | \cdot | denotes the absolute value function .

2 4 0 1 None of the other answers

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Ervin Tronati by Ervin Tronati , 22, Italy

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1 solution

Tom Engelsman
Jan 29, 2017

If ∣ x y ∣ = ∣ x + y ∣ , |xy| = |x + y|, then x y = ± ∣ x + y ∣ ⇒ x + y = ± x y . xy = \pm|x + y| \Rightarrow x + y = \pm xy. Solving for y y in terms of x x produces the system:

y = − x ( 1 ∓ x ) ⇒ y = x x − 1 , − x x + 1 . y = \frac{-x}{(1 \mp x)} \Rightarrow y = \frac{x}{x - 1}, -\frac{x}{x + 1}.

This pair of hyperbolas contains both a vertical & a horizontal asymptote each ⇒ \Rightarrow 2 horiz ( y = ± 1 ) (y = \pm 1) + 2 vert ( x = ± 1 ) (x = \pm 1) = 4 total asymptotes.

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