Question 1

What is the equation, in standard form, of the parabola with focus  (3,0)  and directrix  x=-3  ?

Accepted Answer

$x=1 / 12 y^{2}=0.0833 y$

Question 2

Determine the directrix of the conic with polar form $r=6 /(3-\cos \theta)$

Accepted Answer

x= -6

Question 3

Use the discriminant $B^{2}-4 A C$ to determine if $3 x^{2}-12 x y+4 y^{2}+x-5 y-4=0$  is an ellipse, a parabola, or a hyperbola.

Accepted Answer

96>0 ;  hyperbola

Question 4

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

Accepted Answer

The answer of What type of conic is the graph...

Question 5

Write the parametric equations for the line through the points  (-1,2,4)  and  (6,0,-3) .

Accepted Answer

The answer of Write the parametric equations for the line...

Question 6

$\text { Draw the graph of } r=4 /(2+\cos \theta)$

Accepted Answer

The answer of \(\text { Draw the graph of }...

Question 7

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?

Accepted Answer

The answer of What is the equation of a parabola...

Question 8

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

Accepted Answer

The answer of Find the coordinates of the point ...

Question 9

Find a polar equation for the conic with focus at  (0,0) , eccentricity  4 / 3 , and directrix  y=7 .

Accepted Answer

The answer of Find a polar equation for the conic...

Question 10

What is the vertex of the parabola 
$5 x^{2}-40 x-y+78=0 ?$

Accepted Answer

The answer of What is the vertex of the parabola...

Question 11

For the points  A(-2,4,6), B(0,6,-2)  and  C(4,-8,6) , what is the distance from point  A  to the midpoint of line segment  BC?

Accepted Answer

The answer of For the points  A(-2,4,6), B(0,6,-2) ...

Question 12

Which of the following is the graph of the plane  2 x+6 y+z=1  ?
A) 
B) 
C) 
D)

Accepted Answer

The answer of Which of the following is the graph...

Question 13

Find an equation in standard form for the ellipse whose major axis endpoints are  (-5,2)  and  (3,2)  and whose minor axis has length 6 .

Accepted Answer

The answer of Find an equation in standard form for...

Question 14

After translation of coordinate axes, a quadratic
equation becomes $\frac{\left(x^{\prime}\right)^{2}}{16}+\frac{\left(y^{\prime}\right)^{2}}{9}=1$
had the effect of moving the point  (x, y)=(1,-2)  to
the point $\left(x^{\prime}, y^{\prime}\right)=(0,0)$
equation in  x y  coordinates?
A) $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$
B) $\frac{(x+1)^{2}}{16}+\frac{(y-2)^{2}}{9}=1$
C) $\frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{9}=1$
D) $\frac{\left(x^{\prime}-1\right)^{2}}{16}+\frac{\left(y^{\prime}+2\right)^{2}}{9}=1$

Accepted Answer

The translation of the coordinate axes that moves the point (1, -2) to (0, 0) implies that the original coordinates (x, y) are related to the new coordinates (x', y') by the equations x' = x - 1 and y' = y + 2. Substituting these into the given equation in x'y' coordinates gives the equation in xy coordinates as $\frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{9}=1$.

Question 15

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

Accepted Answer

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

Question 16

Sketch a graph of the equation $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$

Accepted Answer

The answer of Sketch a graph of the equation $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$...

Question 17

Find the asymptotes of the hyperbola $\frac{x^{2}}{81}-\frac{y^{2}}{25}=1$
A) $y=\frac{5}{9} x \text { and } y=-\frac{5}{9} x$
B) $y=\frac{5}{9} x+10.30 \text { and } y=-\frac{5}{9} x+10.30$
C) $y=\frac{9}{5} x \text { and } y=-\frac{9}{5} x$
D)  y=2.06 x  and  y=-2.06 x

Accepted Answer

The asymptotes of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are given by $y = \pm\frac{b}{a}x$. For this hyperbola, $a^2 = 81$ and $b^2 = 25$, so $a = 9$ and $b = 5$, leading to the asymptotes $y = \pm\frac{5}{9}x$.

Question 18

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

Accepted Answer

The answer of Explain how you can use the eccentricity...

Question 19

Find an equation in standard form for the hyperbola centered at  (1,-4) , with one focus at  (7,-4)  and eccentricity  e=2 .

Accepted Answer

The answer of Find an equation in standard form for...

Question 20

Which of the following are the vertices of the hyperbola $\frac{(x-3)^{2}}{25}-\frac{(y+5)^{2}}{169}=1 ?$
A)  (3,8)  and  (3,-18) 
B)  (8,-5)  and  (-2,-5) 
C)  (3,8)  and  (3,-13) 
D)  (16,-5)  and  (-10,-5)

Accepted Answer

The vertices of a hyperbola in the form $\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$ are given by $(h \pm a, k)$ for a horizontal hyperbola. Here, $h=3$, $k=-5$, and $a^{2}=25$, so $a=5$. Thus, the vertices are $(3 \pm 5, -5)$, which simplifies to $(8, -5)$ and $(-2, -5)$.

Quiz 8: Analytic Geometry in Two and Three Dimensions

What is the equation, in standard form, of the parabola with focus (3,0) and directrix x=-3 ?

Determine the directrix of the conic with polar form $r=6 /(3-\cos \theta)$

Use the discriminant $B^{2}-4 A C$ to determine if $3 x^{2}-12 x y+4 y^{2}+x-5 y-4=0$ is an ellipse, a parabola, or a hyperbola.

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

Write the parametric equations for the line through the points (-1,2,4) and (6,0,-3) .

$\text { Draw the graph of } r=4 /(2+\cos \theta)$ $\text { Draw the graph of } r=4 /(2+\cos \theta)$

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?

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 a polar equation for the conic with focus at (0,0) , eccentricity 4 / 3 , and directrix y=7 .

What is the vertex of the parabola $5 x^{2}-40 x-y+78=0 ?$

For the points A(-2,4,6), B(0,6,-2) and C(4,-8,6) , what is the distance from point A to the midpoint of line segment BC?

Which of the following is the graph of the plane 2 x+6 y+z=1 ?

Find an equation in standard form for the ellipse whose major axis endpoints are (-5,2) and (3,2) and whose minor axis has length 6 .

After translation of coordinate axes, a quadratic equation becomes $\frac{\left(x^{\prime}\right) ^{2}}{16}+\frac{\left(y^{\prime}\right) ^{2}}{9}=1$ had the effect of moving the point (x, y) =(1,-2) to the point $\left(x^{\prime}, y^{\prime}\right) =(0,0)$ equation in x y coordinates?

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

Sketch a graph of the equation $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$

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

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

Find an equation in standard form for the hyperbola centered at (1,-4) , with one focus at (7,-4) and eccentricity e=2 .

Which of the following are the vertices of the hyperbola $\frac{(x-3) ^{2}}{25}-\frac{(y+5) ^{2}}{169}=1 ?$

Quiz 8: Analytic Geometry in Two and Three Dimensions

What is the equation, in standard form, of the parabola with focus (3,0) and directrix x=-3 ?

Determine the directrix of the conic with polar form r=6/(3−cos⁡θ)r=6 /(3-\cos \theta)r=6/(3−cosθ)

Use the discriminant B2−4ACB^{2}-4 A CB2−4AC to determine if 3x2−12xy+4y2+x−5y−4=03 x^{2}-12 x y+4 y^{2}+x-5 y-4=03x2−12xy+4y2+x−5y−4=0 is an ellipse, a parabola, or a hyperbola.

What type of conic is the graph of r=3/(1−cos⁡θ)?r=3 /(1-\cos \theta) ?r=3/(1−cosθ)?

Write the parametric equations for the line through the points (-1,2,4) and (6,0,-3) .

Draw the graph of r=4/(2+cos⁡θ)\text { Draw the graph of } r=4 /(2+\cos \theta) Draw the graph of r=4/(2+cosθ)

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?

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

Find a polar equation for the conic with focus at (0,0) , eccentricity 4 / 3 , and directrix y=7 .

What is the vertex of the parabola 5x2−40x−y+78=0?5 x^{2}-40 x-y+78=0 ?5x2−40x−y+78=0?

For the points A(-2,4,6), B(0,6,-2) and C(4,-8,6) , what is the distance from point A to the midpoint of line segment BC?

Which of the following is the graph of the plane 2 x+6 y+z=1 ?

Find an equation in standard form for the ellipse whose major axis endpoints are (-5,2) and (3,2) and whose minor axis has length 6 .

Find the foci of the ellipse (x−2)264+(y+3)225=1\frac{(x-2)^{2}}{64}+\frac{(y+3)^{2}}{25}=164(x−2)2​+25(y+3)2​=1

Sketch a graph of the equation x29+y216=1\frac{x^{2}}{9}+\frac{y^{2}}{16}=19x2​+16y2​=1

Find the asymptotes of the hyperbola x281−y225=1\frac{x^{2}}{81}-\frac{y^{2}}{25}=181x2​−25y2​=1

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

Find an equation in standard form for the hyperbola centered at (1,-4) , with one focus at (7,-4) and eccentricity e=2 .

Which of the following are the vertices of the hyperbola (x−3) 225−(y+5) 2169=1?\frac{(x-3) ^{2}}{25}-\frac{(y+5) ^{2}}{169}=1 ?25(x−3) 2​−169(y+5) 2​=1?

Determine the directrix of the conic with polar form $r=6 /(3-\cos \theta)$

Use the discriminant $B^{2}-4 A C$ to determine if $3 x^{2}-12 x y+4 y^{2}+x-5 y-4=0$ is an ellipse, a parabola, or a hyperbola.

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

$\text { Draw the graph of } r=4 /(2+\cos \theta)$ $\text { Draw the graph of } r=4 /(2+\cos \theta)$

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

What is the vertex of the parabola $5 x^{2}-40 x-y+78=0 ?$

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

Sketch a graph of the equation $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$

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

Which of the following are the vertices of the hyperbola $\frac{(x-3) ^{2}}{25}-\frac{(y+5) ^{2}}{169}=1 ?$