Chapter 10: Q. 84 (page 669)

Prove that the hyperbola
$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$
has the two oblique asymptotes $y = \frac{a}{b} x$ and $y = - \frac{a}{b} x$

Short Answer

Expert verified

When $x \to \pm \infty$ the term $\frac{b^{2}}{x^{2}}$ become zero. Therefore, what remains is $y = \pm \frac{a}{b} x$ which is exactly what you needed to prove.

Step by step solution

Step 1. Given Information

We prove that a hyperbola given with

$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$

has the line

$y = \pm \frac{a}{b} x$

for its asymptotes

Step 2. Concept

We want to express the variable $y$ and then see what value does that expression becomes when we let $x \to \pm \infty$

Step 3. Proving the statement

The first step is to solve the given equation for $y$

$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 \frac{y^{2}}{a^{2}} = 1 + \frac{x^{2}}{b^{2}} y^{2} = a^{2} [1 + \frac{x^{2}}{b^{2}}] y = \pm a \sqrt{1 + \frac{x^{2}}{b^{2}}}$

Second step is to try to get the most similar expression as the wanted one. Therefore, from the previous equation you can get

$y = \pm a \sqrt{\frac{x^{2}}{b^{2}} (\frac{b^{2}}{x^{2}} + 1)}$

and from this

$y = a \frac{x}{b} [\frac{b^{2}}{x^{2}} + 1]$ or $y = \frac{a}{b} x [\frac{b^{2}}{x^{2}} + 1]$

which looks a lot like the wanted term. What is left to do is to observe the value of the term inside the parenthesis when $x \to \pm \infty$

Step 4. Hence Proved

When $x \to \pm \infty$ the term

$\frac{b^{2}}{x^{2}}$

become zero.

Therefore, what remains is

$y = \pm \frac{a}{b} x$

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Prove that the hyperbola
$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$
has the two oblique asymptotes $y = \frac{a}{b} x$ and $y = - \frac{a}{b} x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept

Step 3. Proving the statement

Step 4. Hence Proved

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Prove that the hyperbolay2a2-x2b2=1has the two oblique asymptotesy=abxandy=-abx

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept

Step 3. Proving the statement

Step 4. Hence Proved

One App. One Place for Learning.

Most popular questions from this chapter

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Prove that the hyperbola
$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$
has the two oblique asymptotes $y = \frac{a}{b} x$ and $y = - \frac{a}{b} x$