Solving the task:

asked 2021-08-10

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the “degenerate” conic.\(\displaystyle{x}^{{2}}+{2}{x}{y}+{y}^{{2}}-{1}={0}\)

asked 2020-10-25

Consider the equation:

\(\displaystyle{2}\sqrt{{3}}{x}^{2}-{6}{x}{y}+\sqrt{{3}}{x}+{3}{y}={0}\)

a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola?

b) Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies \(\displaystyle{0}\le\varphi\le\pi\text{/}{2}\))

\(\displaystyle{2}\sqrt{{3}}{x}^{2}-{6}{x}{y}+\sqrt{{3}}{x}+{3}{y}={0}\)

a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola?

b) Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies \(\displaystyle{0}\le\varphi\le\pi\text{/}{2}\))

asked 2021-08-13

Find an equation of the conic described.Graph the equation.

Ellipse; center at (0,0); focus at (0,3); vertex at (0, 5)

Ellipse; center at (0,0); focus at (0,3); vertex at (0, 5)

asked 2021-08-08

Find the equation of the graph for each conic in general form. Identify the conic, the center, the vertex, the co-vertex, the focus (foci), major axis, minor axis, \(\displaystyle{a}^{{2}}\), \(\displaystyle{b}^{{2}}\), and \(\displaystyle{c}^{{2}}\). For hyperbola,find the asymtotes.Sketch the graph

\(\displaystyle{\left({y}+{7}\right)}^{{2}}+{\left({x}-{5}\right)}^{{2}}={25}\)

\(\displaystyle{\left({y}+{7}\right)}^{{2}}+{\left({x}-{5}\right)}^{{2}}={25}\)

asked 2021-08-07

Sketch the following conic sections:

a)\(\displaystyle{y}^{{2}}-{4}{y}+{8}-{4}{x}^{{2}}={8}\)

b)\(\displaystyle{y}^{{2}}-{4}{y}-{4}{x}^{{2}}={8}\)

c)\(\displaystyle{3}{\left({y}-{5}\right)}^{{2}}-{7}{\left({x}+{1}\right)}^{{2}}={1}\)

a)\(\displaystyle{y}^{{2}}-{4}{y}+{8}-{4}{x}^{{2}}={8}\)

b)\(\displaystyle{y}^{{2}}-{4}{y}-{4}{x}^{{2}}={8}\)

c)\(\displaystyle{3}{\left({y}-{5}\right)}^{{2}}-{7}{\left({x}+{1}\right)}^{{2}}={1}\)

asked 2021-02-25

Identify the equation without applying a rotation of axes.

\(\displaystyle{4}{x}^{2}-{8}{x}{y}+{4}{y}^{2}-{3}{x}+{6}={0}\)

\(\displaystyle{4}{x}^{2}-{8}{x}{y}+{4}{y}^{2}-{3}{x}+{6}={0}\)

asked 2021-08-08

Find the largest and smallest distances from the origin to the conic whose equation is \(\displaystyle{5}{x}^{{2}}-{6}{x}{y}+{5}{y}^{{2}}-{32}={0}\) and hence determine the lengths of the semiaxes of this conic.