The Reason for different Forms of Equations

Question

I recently started learning about conic sections and saw people writing the equations for the different figures (circle, parabola, ellipse, and hyperbola) in different forms. (standard form, vertex form, slope-intercept form) My question is what reason there is to write the equations in those different form? Does writing an equation in standard form/vertex form/slope-intercept form give me any additional information?

Answer 1

The reason that there are different forms for writing equations is that they are useful for different things. In some applications, knowing the location of the focci or $y$-intercept may be important. In other applications, making it clear that the curve contains a particular point may be important. It all depends on what the equation will be used for.

Answer 2

It is nice to have various means to represent curves.

I think the implicit equation $$ F(x,y) = x^2 + y^2 - 1 = 0 $$ is more elegant to describe the points of the unit circle than the function $$ y = \pm \sqrt{1-x^2} $$ which needs branches to deal with the multiple values in $y$-direction, but I might use it to plot the graph in Cartesian coordinates.

The parameterized version $$ x = \cos t \quad y = \sin t $$ is useful e.g. to calculate arc lengths. I think that representation has more information than the implicit form, as we declare a direction with increasing parameter.