To calculate: The equation 100y^{2} + 4x=x^{2} + 104 in one of standard forms of the conic sections and identify the conic section.

To calculate: The simplified form of the expression , ${\displaystyle \frac{8}{\sqrt{15}-\sqrt{11}}}$

Evaluate each expression.
P(14,0)

The polar equation of the conic with the given eccentricity and directrix and focus at origin: ${\displaystyle r=41+\cos \theta }$

Decide if the equation defines an ellipse, a hyperbola, a parabola, or no conic section at all.
${\displaystyle \left(a\right)4x^2-9y^2=12\left(b\right)-4x+9y^2=0}$
${\displaystyle \left(c\right)4y^2+9x^2=12\left(d\right)4x^3+9y^3=12}$

Graph the lines and conic sections ${\displaystyle r=\frac{1}{(1+\cos \theta )}}$

Formula for analytical finding ellipse and circle intersection points if exist
I need a formula that will give me all points of random ellipse and circle intersection (ok, not fully random, the center of circle is laying on ellipse curve)
I need step by step solution (algorithm how to find it) if this is possible.

Is there a parametrization of a hyperbola ${\displaystyle x^2-y^2=1}$ by functions x(t) and y(t) both birational?
Consider the hyperbola ${\displaystyle x^2-y^2=1}$. I am aware of some parametrizations like:
1. ${\displaystyle (x\left(t\right),y\left(t\right))=(\frac{t^2+1}{2t},\frac{t^2-1}{2t})}$
2. ${\displaystyle (x\left(t\right),y\left(t\right))=(\frac{t^2+1}{t^2-1},\frac{2t}{t^2-1})}$
3. ${\displaystyle (x\left(t\right),y\left(t\right))=(\text{cosh}t,\text{sinh}t)}$
4. ${\displaystyle (x\left(t\right),y\left(t\right))=(\sec \left(t\right),\tan \left(t\right))}$
The first and the second are by rational functions x(t) and y(t). But the functions are not birational(i.e. with rational inverse of each).
Is there a parametrization where:
- x(t) is rational with inverse also rational, and
- y(t) is rational with inverse also rational?
Is possible, to find a parametrization where both are rational and at least one of the has inverse rational?