Find asymptotes of the curve x^2y -xy^2 +xy+y^2 +x-y=0.
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Answers
Answer:
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Step-by-step explanation:
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Answer:
The asymptotes of given curve are y = 0 and y = x+2.
Step-by-step explanation:
Asymptote of a curve is a line that approaches the graph of a curve without touching it.
Step 1-For finding asymptotes of a curve, we first put y=mx+c at all places of y. Hence,
x²y-xy²+xy+y²+x-y=0
Putting y=mx+c, x²(mx+c)-x(mx+c)²+x(mx+c)+(mx+c)²+x-(mx+c)=0
After opening the brackets, multplying and expanding, we get answer as
mx³+cx²-m²x³-c²x-2mx²c+mx²+cx+m²x²+c²+2mxc+x=0
Step 2- Now we write coefficients of highest power i.e. 3 together and coefficients of the power less than highest power i.e. 2 together and equate them both to 0. Hence,
x³(m-m²)=0 and x²(c-2mc+m+m²)=0
Step 3- Now we compute value of m from the equation of highest power of x by neglecting the variable x and substitute it in the second equation to find value of c. Hence,
m-m²=0 ⇒ m(m-1)=0
We get m=0 or 1.
Substituting it in second equation :
We get c=o for m=0.
For m=1, we get c=2.
Step 4- Finally we put values of m and c in the equation y=mx+c, we get the desired asymptotes.
Therefore, asymptotes are y=0 and y=x+2.