91Ó°ÊÓ

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. $$ r=6 \cot \theta $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=5 $$

Find an equation for the set of points in an \(x y\)-plane such that the difference of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$F(13,0), \quad F^{\prime}(-13,0) ; \quad k=24$$

A searchlight reflector has the shape of a paraboloid, with the light source at the focus. If the reflector is 3 feet across at the opening and 1 foot deep, where is the focus?

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. $$ y=2-7 \sqrt{1-\frac{(x+1)^{2}}{9}} $$

Find an equation for the set of points in an \(x y\)-plane such that the difference of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$F(5,0), \quad F^{\prime}(-5,0) ; \quad k=8$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=-2 $$

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. $$ y=-1+\sqrt{1-\frac{(x-3)^{2}}{16}} $$

Exer. 45-78: Sketch the graph of the polar equation. $$ \theta=-\pi / 6 $$

A sound receiving dish used at outdoor sporting events is constructed in the shape of a paraboloid, with its focus 5 inches from the vertex. Determine the width of the dish if the depth is to be 2 feet.