Use concept of tracing, draw the curve y^2(x^2 + y^2) =a^2(y^2-x^2)
write symmetry origin assymptode region
Answers
We want to trace the curve y2=x3.
Symmetry:
The curve is symmetric about the X axis.
Domain and Range:
The curve has two branches. It can be considered as the union of two functions: y=x3/2 and y=−x3/2.
The domain is the set of non-negative real numbers for both the functions and the range is the set of non-negative real numbers for the first function and the set of non-positive real numbers for the second function.
As x tends to ∞, one branch tends to ∞ and the other branch tends to −∞.
As x tends to 0, both the branches tend to 0.
Asymptotes
The curve does not have any asymptotes.
Intersection with the coordinate axes:
Both the branches touch the origin. It does not touch or intersect the coordinate axes at any other point.
The curve forms a cusp at the origin i.e. both the branches have the same tangent a the origin. The common tangent is y=0.
Derivatives:
The derivative of the curve with respect to x is 32x−−√ for one branch and −32x−−√ for the other branch.
⇒ One branch is monotonically increasing and the other branch is monotonically decreasing.
The curve does not have any local or global maxima or minima.
A table can now be made giving values of the Y coordinate for various values of the X coordinate for both the branches of the curve and the curve can be plotted