Exam 8: Analytic Geometry in Two and Three Dimensions

Find the coordinates of the point P(x, y)=(3,4) in the coordinate system x' y' obtained by a rotation of $\alpha=\pi / 3 .$

Find the foci of the ellipse $\frac{(x-2)^{2}}{64}+\frac{(y+3)^{2}}{25}=1$

In three-dimensional space, what is the equation of the set of points that are 8 units from the point (-3,5,2)?

Identify the conic without drawing its graph. State the eccentricity and directrix. $r=\frac{20}{7-4 \sin \theta}$

Find the asymptotes of the hyperbola $\frac{x^{2}}{81}-\frac{y^{2}}{25}=1$

Find the vertex of the parabola $y=2 x^{2}-12 x+23$

What type of conic is the graph of $r=3 /(1-\cos \theta) ?$

Explain how you can use the eccentricity to determine whether the graph of a given quadratic equation is an ellipse or a circle.

What is the equation of a parabola with vertex (-4,-4) that passes through (0,0) and has its axis of symmetry parallel to the y -axis?

What is the magnitude of the vector $\mathbf{v}=\langle 1,-4,2\rangle ?$

For the hyperbola $\frac{(y-5)^{2}}{9}-\frac{(x-3)^{2}}{25}=1$ find the center, foci, endpoints of the transverse axis, and the equations of the asymptotes.

Identify the type of conic given by the equation $2 x^{2}-x y-3 y^{2}-2 x+4 y-6=0$