Curve Tracing f(x,y)=0
Symmetry #
- If f(-x,y)=f(x,y) the curve is symmetric with respect to y axis. (Even degree of x)
- If f(x,-y)=f(x,y): the curve is symmetric with respect to x axis. (Even degree of y)
Origin #
Passes through the origin if f(0,0)=0
Tangent at origin #
Equating least degree term to 0, we can get the tangent at the origin.
dy/dx at any point on f(x,y)=0 #
$$ \begin{align} f(x,y)=0\\ \frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}\cdot \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y}\cdot \frac{\partial y}{\partial t}\\ \delta f=0=\frac{\partial f}{\partial x}\delta x+\frac{\partial f}{\partial y}\delta y\\ \frac{dy}{dx}=-\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} \end{align} $$
Intersection with coordinate axes #
- f(xâ‚€,0)=0: intersection with x-axis (y=0)
- f(0,yâ‚€)=0: intersection with y-axis (x=0)
Asymptotes #
- To find the ones parallel to x-axis: Equate the coefficient of the term containing the highest degree in x to 0
- To find the ones parallel to y-axis: Equate the coefficient of the term containing the highest degree in y to 0
Region of existence #
Can be found by seeing the possible values of (x,y) that f(x,y)=0 satisfies.
EXAMPLES #
http://www.dspmuranchi.ac.in/pdf/Blog/Integration%20113.pdf