91Ó°ÊÓ

In Exercises 49–56, identify each equation without completing the square. $$ 9 x^{2}+4 y^{2}-36 x+8 y+31=0 $$

Prove that the polar equation of a planet's elliptical orbit is $$ r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta} $$ where \(e\) is the eccentricity and \(2 a\) is the length of the major axis.

In Exercises \(51-60,\) convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$ x^{2}+4 y^{2}+10 x-8 y+13=0 $$

Find two different sets of parametric equations for each rectangular equation. \(y=2 x-5\)

graph each relation. Use the relation’s graph to determine its domain and range. $$ \frac{y^{2}}{16}-\frac{x^{2}}{9}=1 $$

Find two different sets of parametric equations for each rectangular equation. \(y=x^{2}+4\)

In Exercises 49–56, identify each equation without completing the square. $$ 100 x^{2}-7 y^{2}+90 y-368=0 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can verify that \(2 x y-9=0\) is the equation of a hyperbola by rotating the axes through \(45^{\circ}\) or by showing that \(B^{2}-4 A C>0\)

Expand: $$ \log _{b}\left(x^{3} \sqrt{y}\right) $$

In Exercises \(51-60,\) convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$ 4 x^{2}+y^{2}+16 x-6 y-39=0 $$