91Ó°ÊÓ

Let an \(x^{\prime} y^{\prime}\) -coordinate system be obtained by rotating an \(x y\) -coordinate system through an angle of \(30^{\circ} .\) Use ( 5) to find an equation in \(x^{\prime} y^{\prime}\) -coordinates of the curve \(y=x^{2}\).

Find parametric equations for the curve, and check your work by generating the curve with a graphing utility. The portion of the circle \(x^{2}+y^{2}=1\) that lies in the third quadrant, oriented counterclockwise.

Prove that a hyperbola is an equilateral hyperbola if and only if \(e=\sqrt{2}\).

Sketch the hyperbola, and label the vertices, foci, and asymptotes. $$ \begin{array}{l}{\text { (a) } x^{2}-4 y^{2}+2 x+8 y-7=0} \\ {\text { (b) } 16 x^{2}-y^{2}-32 x-6 y=57}\end{array} $$

Sketch the polar curve and find polar equations of the tangent lines to the curve at the pole. $$ r=4 \sin \theta $$

Sketch the polar curve and find polar equations of the tangent lines to the curve at the pole. $$ r=4 \sqrt{\cos 2 \theta} $$

Let an \(x^{\prime} y^{\prime}\) -coordinate system be obtained by rotating an \(x y\) -coordinate system through an angle \(\theta .\) Prove: For every value of \(\theta,\) the equation \(x^{2}+y^{2}=r^{2}\) becomes the equation \(x^{\prime 2}+y^{\prime 2}=r^{2} .\) Give a geometric explanation.

Find parametric equations for the curve, and check your work by generating the curve with a graphing utility. A vertical line intersecting the \(x\) -axis at \(x=2,\) oriented upward.

Find an equation for the parabola that satisfies the given conditions. $$ \begin{array}{l}{\text { (a) Vertex }(0,0) ; \text { focus }(3,0)} \\ {\text { (b) Vertex }(0,0) ; \text { directrix } y=\frac{1}{4}}\end{array} $$

Find parametric equations for the curve, and check your work by generating the curve with a graphing utility. The ellipse \(x^{2} / 4+y^{2} / 9=1,\) oriented counterclockwise.