Question

(x/2) squared - (y/2) squared=1
find the asymptotes

Answer 1

Let’s first consider something easier.

x^2 – y^2 = 1.
This will trace out a hyperbola.
The asymptotes are lines that the hyperbola will approach as x goes to infinity.

Asymptotes are lines.
So we want equations of lines.
This suggests we solve for y in terms of x in the hyperbola, and
then see what happens to y as x goes to infinity.

Our hyperbola is x^2 – y^2 = 1. So y^2 = x^2 – 1.

So y = sqrt(x^2 – 1) or y = - sqrt(x^2 – 1).

Now, we see that when x goes to infinity,
the expression under the square root goes to infinity.
So y = sqrt(x^2 – 1) goes off to +infinity and y = - sqrt(x^2 – 1) goes off to – infinity.

But what line does it approach?
As x becomes large, x^2 – 1 is approximately just x^2.
So then y is approximately sqrt(x^2) = x. Therefore, y = x is an asymptote.

Similarly, for negative values of y,
y = - sqrt(x^2 – 1) means that y is approximately –sqrt(x^2) = -x.

So y = x and y = -x are the asymptotes.

In general, if your hyperbola is (x/a)^2 – (y/b)^2 = 1, then (y/b)^2 = (x/a)^2 – 1.
So y/b = sqrt( (x/a)^2 – 1) or y/b = - sqrt((x/a)^2 – 1).

So as x goes to infinity, (x/a)^2 – 1 is approximately just (x/a)^2 since subtracting 1 becomes insignificant.
So then y/b is approximately sqrt( (x/a)^2) = x/a. Therefore, y ~ (b/a) * x.
Similarly for the negative values of y, y ~ -(b/a) * x.

These are your asymptotes for the hyperbola (x/a)^2 – (y/b)^2 = 1.

You can also use the same chain of reasoning to find the asymptotes for
the hyperbola (y/b)^2 – (x/a)^2 = 1 if you needed to.