91Ó°ÊÓ

A circle \(C\) of radius \(b\) rolls on the outside of the circle \(x^{2}+y^{2}=a^{2}\), and \(b

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r\left(r \sin ^{2} \theta-\cos \theta\right)=3 $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ y=-6 \sqrt{1-\frac{x^{2}}{25}} $$

Exer. 39-40: Find an equation for the parabola that has a vertical axis and passes through the given points. $$ P(3,-1), \quad Q(1,-7), \quad R(-2,14) $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x+4=y^{2}+y$$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x^{2}+3 x=3 y-6$$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ x=-\frac{1}{3} \sqrt{9-y^{2}} $$

Exer. 41-42: Find an equation for the parabola that has a horizontal axis and passes through the given points. $$ P(-1,1), \quad Q(11,-2), \quad R(5,-1) $$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=8 \sin \theta-2 \cos \theta $$

Exer. 41-42: Find an equation for the parabola that has a horizontal axis and passes through the given points. $$ P(2,1), \quad Q(6,2), \quad R(12,-1) $$