Exam 8: Analytic Geometry in Two and Three Dimensions

Solve the problem. -A rectangular board is 14 by 20 units. How far from the short side of the board will the foci be located to determine the largest elliptical tabletop?

A quadratic equation has been transformed using the rotation equations x = 0.8u - 0.6v and y = 0.6u + 0.8v. Convert the given point from (u, v) coordinates back to (x, y) coordinates. -hyperbola foci $( \pm 3,0 )$

Find an equation that matches the parabola's graph. -

Graph the ellipse. - $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$

Find the values of e, a, b, and c. - $r = \frac { 24 } { 4 + 2 \sin \theta }$

Find the vertices and foci of the hyperbola. - $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 12 } = 1$

Match the polar equation with its graph. - $r = \frac { 8 } { 4 - \sin \theta }$

Find the eccentricity of the ellipse. - $x ^ { 2 } + 3 y ^ { 2 } = 15$

Solve the problem. -A railroad tunnel is shaped like a semiellipse. The height of the tunnel at the center is $28 \mathrm { ft }$ and the vertical clearance must be $26 \mathrm { ft }$ at a point $18 \mathrm { ft }$ from the center. Find an equation for the ellipse.

Find the standard form of the equation of the parabola. -Focus at $( 0,3 )$ , directrix $y = - 3$

Find the eccentricity of the hyperbola. - $x ^ { 2 } - 2 y ^ { 2 } = 6$

Compute the distance between the points. - $( 4,4 , - 6 ) , ( - 4,5,10 )$

Match the polar equation with its graph. - $r = \frac { 12 } { 8 + 6 \cos \theta }$

Graph the ellipse. - $25 x^{2}+25 y^{2}=625$

Find parametric equations for the line described below. -The line through the point $P ( - 3,4 , - 2 )$ in the direction of the vector $\langle - 8,4 , - 5 \rangle$

Match the given graph with its equation. -

Graph the hyperbola. - $25 x^{2}-4 y^{2}=100$

Compute the distance between the points. - $( 6 , - 2,9 ) , ( x , y , z )$

Find the midpoint of the segment PQ. - $\mathrm { P } ( - 9,10 , - 3 ) , \mathrm { Q } ( 1,15,7 )$

Evaluate the expression. - $\mathbf { r } = \langle 7 , - 5 , - 3 \rangle , \mathbf { v } = \langle 2,4 , - 5 \rangle , \mathbf { w } = \langle 1,7,5 \rangle$ $\mathbf { v } \cdot \mathbf { w }$