The exercise provides the equation for a degenerate conical section. Fill in the square and describe the graph of each equation. \(\displaystyle{4}{x^2}-{y^2}-{32}{x}-{4}{y}+{60}={0}\)

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

The type of the conic section using the Discriminant Test and plot the curve using a computer algebra system. \(\displaystyle{x}^{2}\ -\ {2}{x}{y}\ +\ {y}^{2}\ +\ {24}{x}\ -\ {8}={0}\)

To determine: The conic section and to find the vertices and foci: \(\displaystyle{x}^{2}\ -\ {y}^{2}{y}={4}\)

Use the Discriminant Test to determine the type of the conic section (in every case non degenerate equation). Plot the curve if you have a computer algebra system. \(\displaystyle{2}{x}^{2}-{8}{x}{y}+{3}{y}^{2}-{4}={0}\)

Find out what kind of conic section the following quadratic form represents and transform it to principal axes. Express \(\displaystyle\vec{{x}}^{T}={\left[{x}_{{1}}{x}_{{2}}\right]}\) in terms of the new coordinate vector \(\displaystyle\vec{{y}}^{T}={\left[{y}_{{1}}{y}_{{2}}\right]}\) \(\displaystyle{{x}_{{1}}^{{2}}}-{12}{x}_{{1}}{x}_{{2}}+{{x}_{{2}}^{{2}}}={70}\)

We need to verify theorem 5 Focus-Directrix definition: \(\displaystyle{0}\ {<}\ {e}\ {<}\ {1}\ \text{and}\ {c}={d}{\left({e}\ -\ {2}\ -\ {1}\right)}\)

To find: The equation of the given hyperbola. The foci of the hyperbola is \(\displaystyle{\left({0},\ \pm\ {5}\right)}{\quad\text{and}\quad} \text{ entricity is }{e} -{1.5}\).