Find The asymptotes of the curve (a^2/x^2)+(b^2/y^2)=1

Share

Find The asymptotes of the curve

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

solution

\frac{a^2}{x^2}+\frac{b^2}{y^2}=1 \Rightarrow \,b^2x^2+a^2y^2=x^2y^2

equating the coefficient of highest powers of x to zero ,

we\,get\, \;b^2-y^2=0\Rightarrow\;y=\pm b

equating the coefficient of highest powers of y to zero ,

we\,get\, \;a^2-x^2=0\Rightarrow\;x=\pm a
\therefore x=\pm \;a \; and \;y=\pm \;b \; are\; the \;required\;asymptotes 

Q-Find out circular asymptotes of the curve

r=\frac{3\theta}{\theta+1}

Solution – the equation of the curve is r=\frac{3\theta}{\theta+1}=f(\theta)\, ,say

\lim_{\theta\rightarrow \infty}f(\theta)=\lim_{\theta\rightarrow \infty}\frac{3\theta}{\theta+1 }=\lim_{\theta\rightarrow \infty}\frac{3}{1+\frac{1}{\theta} }=3

r=3 circular asymptote


Find the asymptotes parallel to the axes of the curve

x^2 y^2 - x^2 - y^2 - x - y + 1 = 0

Given curve is

x^2 y^2 - x^2 - y^2 - x - y + 1 = 0

Given curve is of degree 4 so we cannot have more than 4 asymptotes 

Related  Find the values of x for which the series x + x^3 + x^5 + • • • converges, and express the sum as a function of x. infinite series problems and solutions

Asymptotes parallel to x axis :

Equating the coefficient of highest power of x (i.e,\; of\; x^2 \;) equals to zero,

We get

y^2-1=0  \implies y^2=1 \implies y =\pm1 \\ \implies y-1=0 \; and \; y+1=0

Asymptotes parallel to y axis :

Equating the coefficient of highest power of y (i.e,\; of\; y^2 \;) equals to zero,

We get

x^2-1=0   \implies (x-1)(x+1)=0 \\ \implies x-1=0 \; and \; x+1=0

The asymptotes of the given curve are

{\color{Blue} y-1=0,\;y+1=0,\; x-1=0 \; and \; x+1=0}

keywords

how to find asymptotes of a curve

asymptotes of a curve


Share

Leave a Reply

Your email address will not be published. Required fields are marked *

Top 5 Most Expensive Domains Ever Sold 4 Must-Try ChatGPT Alternatives: Perplexity AI, BardAI, Pi, and More! Types of Trading Techniques in the Stock Market. ChatGPT app now available in India this AI chatbot can help you make your life more productive. What is wrong with following function code?