Question 1

Graph the hyperbola.
-$25 y^{2}-9 x^{2}=225$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 2

Determine the eccentricity, the type of conic, and the directrix.
-$$r = \frac { 7 } { 9 - 6 \cos \theta }$$

Accepted Answer

A)  Eccentricity $= \frac { 2 } { 3 }$; ellipse; directrix $x = - \frac { 7 } { 6 }$ 
B)  Eccentricity $= \frac { 2 } { 3 } ;$ hyperbola; directrix $x = - \frac { 21 } { 2 }$ 
C)  Eccentricity $= \frac { 3 } { 2 }$; ellipse; directrix $x = - \frac { 7 } { 6 }$ 
D)  Eccentricity $= \frac { 6 } { 7 }$; hyperbola; directrix $y = - \frac { 14 } { 3 }$ 
A)  Eccentricity $= \frac { 2 } { 3 }$; ellipse; directrix $x = - \frac { 7 } { 6 }$ 
B)  Eccentricity $= \frac { 2 } { 3 } ;$ hyperbola; directrix $x = - \frac { 21 } { 2 }$ 
C)  Eccentricity $= \frac { 3 } { 2 }$; ellipse; directrix $x = - \frac { 7 } { 6 }$ 
D)  Eccentricity $= \frac { 6 } { 7 }$; hyperbola; directrix $y = - \frac { 14 } { 3 }$

Question 3

Find an equation in standard form for the ellipse that satisfies the given conditions.
-An ellipse with foci at $( 2,0 )$ and $( 2 , - 6 )$; major axis length of 10

Accepted Answer

A)  $\frac { ( y + 3 ) ^ { 2 } } { 25 } + \frac { ( x - 2 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( y + 3 ) ^ { 2 } } { 25 } + \frac { ( x + 2 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x + 3 ) ^ { 2 } } { 16 } + \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x + 3 ) ^ { 2 } } { 25 } + \frac { ( y + 2 ) ^ { 2 } } { 16 } = 1$ 
A)  $\frac { ( y + 3 ) ^ { 2 } } { 25 } + \frac { ( x - 2 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( y + 3 ) ^ { 2 } } { 25 } + \frac { ( x + 2 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x + 3 ) ^ { 2 } } { 16 } + \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x + 3 ) ^ { 2 } } { 25 } + \frac { ( y + 2 ) ^ { 2 } } { 16 } = 1$

Question 4

Find an equation in standard form for the hyperbola that satisfies the given conditions.
-Center $( 4 , - 5 )$, focus $( 11 , - 5 )$, vertex $( 7 , - 5 )$

Accepted Answer

A)  $\frac { ( x - 4 ) ^ { 2 } } { 9 } - \frac { ( y + 5 ) ^ { 2 } } { 49 } = 1$ 
B)  $\frac { ( x - 4 ) ^ { 2 } } { 9 } - \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { 9 } - \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { 40 } - \frac { ( y + 5 ) ^ { 2 } } { 9 } = 1$ 
A)  $\frac { ( x - 4 ) ^ { 2 } } { 9 } - \frac { ( y + 5 ) ^ { 2 } } { 49 } = 1$ 
B)  $\frac { ( x - 4 ) ^ { 2 } } { 9 } - \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { 9 } - \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { 40 } - \frac { ( y + 5 ) ^ { 2 } } { 9 } = 1$

Question 5

Use the given substitutions to convert the equation to a (u, v) equation with no cross-product term.
-$7 \mathrm { x } ^ { 2 } + 5 \sqrt { 3 } \mathrm { xy } + 2 \mathrm { y } ^ { 2 } - 31 = 0 ;$ use $x = \sqrt { 3 } \mathrm { u } / 2 - \mathrm { v } / 2$ and $\mathrm { y } = \mathrm { u } / 2 + \sqrt { 3 } \mathrm { v } / 2$

Accepted Answer

A)  $\frac { 19 } { 2 } u ^ { 2 } - 2 v ^ { 2 } = 1$ 
B)  $\frac { 19 } { 62 } \mathrm { u } ^ { 2 } + \frac { 1 } { 62 } \mathrm { v } ^ { 2 } = 1$ 
C)  $\frac { 19 } { 62 } u ^ { 2 } - \frac { 1 } { 62 } v ^ { 2 } = 1$ 
D)  $\frac { 19 } { 2 } u ^ { 2 } - \frac { 1 } { 62 } v ^ { 2 } = - 1$ 
A)  $\frac { 19 } { 2 } u ^ { 2 } - 2 v ^ { 2 } = 1$ 
B)  $\frac { 19 } { 62 } \mathrm { u } ^ { 2 } + \frac { 1 } { 62 } \mathrm { v } ^ { 2 } = 1$ 
C)  $\frac { 19 } { 62 } u ^ { 2 } - \frac { 1 } { 62 } v ^ { 2 } = 1$ 
D)  $\frac { 19 } { 2 } u ^ { 2 } - \frac { 1 } { 62 } v ^ { 2 } = - 1$

Question 6

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
-$$9 x ^ { 2 } - 6 x y + 6 y ^ { 2 } - 20 = 0$$

Accepted Answer

A)  Hyperbola 
B)  Ellipse 
C)  Parabola 
A)  Hyperbola 
B)  Ellipse 
C)  Parabola

Question 7

Solve the problem.
-The roof of a building is in the shape of the hyperbola $y ^ { 2 } - x ^ { 2 } = 11$, where $x$ and $y$ are in meters. Refer to the figure and determine the height $h$ of the outside walls.
 
$a=b=4 m$

Accepted Answer

A)  $27 \mathrm {~m}$ 
B)  $7 \mathrm {~m}$ 
C)  $5.2 \mathrm {~m}$ 
D)  $5 \mathrm {~m}$ 
A)  $27 \mathrm {~m}$ 
B)  $7 \mathrm {~m}$ 
C)  $5.2 \mathrm {~m}$ 
D)  $5 \mathrm {~m}$

Question 8

Find the vertex, focus, directrix, and focal width of the parabola.
-$$\mathrm { x } ^ { 2 } = 36 \mathrm { y }$$

Accepted Answer

A)  Vertex: $( 0,0 )$; Focus: $( 9,0 )$; Directrix: $x = 9$; Focal width: 9 
B)  Vertex: $( 0,0 )$; Focus: $( 0,9 )$; Directrix: $y = - 9$; Focal width: 36 
C)  Vertex: $( 0,0 )$; Focus: $( 9,0 )$; Directrix: $y = 9$; Focal width: 144 
D)  Vertex: $( 0,0 )$; Focus: $( 0 , - 9 )$; Directrix: $x = - 9$; Focal width: 144 
A)  Vertex: $( 0,0 )$; Focus: $( 9,0 )$; Directrix: $x = 9$; Focal width: 9 
B)  Vertex: $( 0,0 )$; Focus: $( 0,9 )$; Directrix: $y = - 9$; Focal width: 36 
C)  Vertex: $( 0,0 )$; Focus: $( 9,0 )$; Directrix: $y = 9$; Focal width: 144 
D)  Vertex: $( 0,0 )$; Focus: $( 0 , - 9 )$; Directrix: $x = - 9$; Focal width: 144

Question 9

Find the vertex, the focus, and the directrix of the parabola.
-$$y ^ { 2 } + 4 x + 8 y + 20 = 0$$

Accepted Answer

A)  Vertex: $( - 1 , - 4 )$; Focus: $( - 2 , - 4 )$; Directrix: $x = 0$ 
B)  Vertex: $\left( - \frac { 1 } { 2 } , - 1 \right)$; Focus: $( 0 , - 1 )$; Directrix: $y = - 2$ 
C)  Vertex: $( - 2 , - 4 )$; Focus: $\left( - \frac { 5 } { 4 } , - 4 \right)$; Directrix: $y = - \frac { 3 } { 4 }$ 
D)  Vertex: $( - 3 , - 1 )$; Focus: $( - 5 , - 1 )$; Directrix: $y = - 3$ 
A)  Vertex: $( - 1 , - 4 )$; Focus: $( - 2 , - 4 )$; Directrix: $x = 0$ 
B)  Vertex: $\left( - \frac { 1 } { 2 } , - 1 \right)$; Focus: $( 0 , - 1 )$; Directrix: $y = - 2$ 
C)  Vertex: $( - 2 , - 4 )$; Focus: $\left( - \frac { 5 } { 4 } , - 4 \right)$; Directrix: $y = - \frac { 3 } { 4 }$ 
D)  Vertex: $( - 3 , - 1 )$; Focus: $( - 5 , - 1 )$; Directrix: $y = - 3$

Question 10

Find the center, vertices, and foci of the ellipse with the given equation.
-$$\frac { x ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 625 } = 1$$

Accepted Answer

A)  Center: $( 0,0 )$; Vertices: $( - 25,0 ) , ( 25,0 )$; Foci: $( - 20,0 ) , ( 20,0 )$ 
B)  Center: $( 0,0 )$; Vertices: $( - 25,0 ) , ( 25,0 )$; Foci: $( 0 , - 15 ) , ( 0,15 )$ 
C)  Center: $( 0,0 )$; Vertices: $( 0 , - 25 ) , ( 0,25 )$; Foci: $( 0 , - 20 ) , ( 0,20 )$ 
D)  Center: $( 0,0 )$; Vertices: $( 0 , - 25 ) , ( 0,25 )$; Foci: $( - 15,0 ) , ( 15,0 )$ 
A)  Center: $( 0,0 )$; Vertices: $( - 25,0 ) , ( 25,0 )$; Foci: $( - 20,0 ) , ( 20,0 )$ 
B)  Center: $( 0,0 )$; Vertices: $( - 25,0 ) , ( 25,0 )$; Foci: $( 0 , - 15 ) , ( 0,15 )$ 
C)  Center: $( 0,0 )$; Vertices: $( 0 , - 25 ) , ( 0,25 )$; Foci: $( 0 , - 20 ) , ( 0,20 )$ 
D)  Center: $( 0,0 )$; Vertices: $( 0 , - 25 ) , ( 0,25 )$; Foci: $( - 15,0 ) , ( 15,0 )$

Question 11

Solve the problem.
-The roof of a building is in the shape of the hyperbola $8 y ^ { 2 } - x ^ { 2 } = 22$, where $x$ and $y$ are in meters. Refer to the figure and determine the height, h, of the outside walls.

$a = b = 6 m$

Accepted Answer

A)  $7.6 \mathrm {~m}$ 
B)  $2.7 \mathrm {~m}$ 
C)  $58 \mathrm {~m}$ 
D)  $7.3 \mathrm {~m}$ 
A)  $7.6 \mathrm {~m}$ 
B)  $2.7 \mathrm {~m}$ 
C)  $58 \mathrm {~m}$ 
D)  $7.3 \mathrm {~m}$

Question 12

Find the linear equations that can be used to convert an (x, y) equation to a (u, v) equation using the given angle of
rotation
-$$\theta = 80 ^ { \circ }$$

Accepted Answer

A)  $x = 0.985 \mathrm { u } - 0.174 \mathrm { v }$
 $y = 0.174 u + 0.985 v$ 
B)  $x = 0.174 u - 0.985 \mathrm { v }$
$y = 0.174 u + 0.985 v$ 
C)  $x = 0.985 \mathrm { u } + 0.174 \mathrm { v }$
 $y = 0.985 u - 0.174 v$ 
D)  $x = 0.174 u - 0.985 v$ 
 $y = 0.985 u + 0.174 v$ 
A)  $x = 0.985 \mathrm { u } - 0.174 \mathrm { v }$
 $y = 0.174 u + 0.985 v$ 
B)  $x = 0.174 u - 0.985 \mathrm { v }$
$y = 0.174 u + 0.985 v$ 
C)  $x = 0.985 \mathrm { u } + 0.174 \mathrm { v }$
 $y = 0.985 u - 0.174 v$ 
D)  $x = 0.174 u - 0.985 v$ 
 $y = 0.985 u + 0.174 v$

Question 13

Solve the problem.
-Find the vector from $A$ to the midpoint of $B C$.
$$\mathrm { A } = ( 5,8,1 ) , \mathrm { B } = ( 2,7,6 ) , \mathrm { C } = ( - 9,7,4 )$$

Accepted Answer

A)  $\langle - 5.5 , - 1,4 \rangle$ 
B)  $\langle - 8.5 , - 1,4 \rangle$ 
C)  $\langle 8.5,1 , - 8.5 \rangle$ 
D)  $\langle - 10 , - 56 , - 6 \rangle$ 
A)  $\langle - 5.5 , - 1,4 \rangle$ 
B)  $\langle - 8.5 , - 1,4 \rangle$ 
C)  $\langle 8.5,1 , - 8.5 \rangle$ 
D)  $\langle - 10 , - 56 , - 6 \rangle$

Question 14

Use axis of rotation formulas for x and y to transform the quadratic equation to an equation in (u, v) coordinates with no
cross-product term.
-$7 x ^ { 2 } + 6 \sqrt { 3 } x y + 13 y ^ { 2 } = 57$

Accepted Answer

A)  $\frac { 16 u ^ { 2 } } { 57 } + \frac { 4 v ^ { 2 } } { 57 } = 1$ 
B)  $\mathrm { v } = \frac { \mathrm { u } ^ { 2 } } { 4 }$ 
C)  $\frac { 4 u ^ { 2 } } { 57 } + \frac { 16 v ^ { 2 } } { 57 } = 1$ 
D)  $\frac { 3 \mathrm { u } ^ { 2 } } { 114 } + \frac { \mathrm { v } ^ { 2 } } { 114 } = 1$ 
A)  $\frac { 16 u ^ { 2 } } { 57 } + \frac { 4 v ^ { 2 } } { 57 } = 1$ 
B)  $\mathrm { v } = \frac { \mathrm { u } ^ { 2 } } { 4 }$ 
C)  $\frac { 4 u ^ { 2 } } { 57 } + \frac { 16 v ^ { 2 } } { 57 } = 1$ 
D)  $\frac { 3 \mathrm { u } ^ { 2 } } { 114 } + \frac { \mathrm { v } ^ { 2 } } { 114 } = 1$

Question 15

Find an equation in standard form for the ellipse that satisfies the given conditions.
-The horizontal major axis is of length 18 , and the minor axis is of length 10 .

Accepted Answer

A)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 81 } = 1$ 
B)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 25 } = 1$ 
D)  $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 9 } = 1$ 
A)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 81 } = 1$ 
B)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 25 } = 1$ 
D)  $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 9 } = 1$

Question 16

Match the given graph with its equation.
-

Accepted Answer

A)  $\frac { ( y + 3 ) ^ { 2 } } { 4 } - \frac { ( x - 1 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( x - 1 ) ^ { 2 } } { 4 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$ 
D)  $\frac { ( x + 3 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 4 } = 1$ 
A)  $\frac { ( y + 3 ) ^ { 2 } } { 4 } - \frac { ( x - 1 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( x - 1 ) ^ { 2 } } { 4 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$ 
D)  $\frac { ( x + 3 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 4 } = 1$

Question 17

Graph the parabola.
-$$( y - 8 ) ^ { 2 } = 6 ( x + 2 )$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 18

Find the standard form of the equation of the parabola.
-Vertex at the origin, opens to the right, focal width $= 6$

Accepted Answer

A)  $y ^ { 2 } = 6 x$ 
B)  $y ^ { 2 } = 1.5 x$ 
C)  $x ^ { 2 } = 6 y$ 
D)  $y ^ { 2 } = - 6 x$ 
A)  $y ^ { 2 } = 6 x$ 
B)  $y ^ { 2 } = 1.5 x$ 
C)  $x ^ { 2 } = 6 y$ 
D)  $y ^ { 2 } = - 6 x$

Question 19

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
-$$x ^ { 2 } + x y + y ^ { 2 } + 4 x - 3 y - 3 = 0$$

Accepted Answer

A)  Hyperbola 
B)  Ellipse 
C)  Parabola 
A)  Hyperbola 
B)  Ellipse 
C)  Parabola

Question 20

Find the standard form of the equation of the parabola.
-Focus at $( 9 , - 5 )$, directrix $y = - 15$

Accepted Answer

A)  $( x - 9 ) ^ { 2 } = 20 ( y + 5 )$ 
B)  $( y + 5 ) ^ { 2 } = 20 ( x - 9 )$ 
C)  $( x - 9 ) ^ { 2 } = 20 ( y + 10 )$ 
D)  $( x + 5 ) ^ { 2 } = 20 ( y + 10 )$ 
A)  $( x - 9 ) ^ { 2 } = 20 ( y + 5 )$ 
B)  $( y + 5 ) ^ { 2 } = 20 ( x - 9 )$ 
C)  $( x - 9 ) ^ { 2 } = 20 ( y + 10 )$ 
D)  $( x + 5 ) ^ { 2 } = 20 ( y + 10 )$

Graph the hyperbola. - $25 y^{2}-9 x^{2}=225$ $Graph the hyperbola. - 25 y^{2}-9 x^{2}=225$

Determine the eccentricity, the type of conic, and the directrix. - $r = \frac { 7 } { 9 - 6 \cos \theta }$

Find an equation in standard form for the ellipse that satisfies the given conditions. -An ellipse with foci at $( 2,0 )$ and $( 2 , - 6 )$ ; major axis length of 10

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Center $( 4 , - 5 )$ , focus $( 11 , - 5 )$ , vertex $( 7 , - 5 )$

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $9 x ^ { 2 } - 6 x y + 6 y ^ { 2 } - 20 = 0$

Find the vertex, focus, directrix, and focal width of the parabola. - $\mathrm { x } ^ { 2 } = 36 \mathrm { y }$

Find the vertex, the focus, and the directrix of the parabola. - $y ^ { 2 } + 4 x + 8 y + 20 = 0$

Find the center, vertices, and foci of the ellipse with the given equation. - $\frac { x ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 625 } = 1$

Find the linear equations that can be used to convert an (x, y) equation to a (u, v) equation using the given angle of rotation - $\theta = 80 ^ { \circ }$

Solve the problem. -Find the vector from $A$ to the midpoint of $B C$ . $\mathrm { A } = ( 5,8,1 ) , \mathrm { B } = ( 2,7,6 ) , \mathrm { C } = ( - 9,7,4 )$

Use axis of rotation formulas for x and y to transform the quadratic equation to an equation in (u, v) coordinates with no cross-product term. - $7 x ^ { 2 } + 6 \sqrt { 3 } x y + 13 y ^ { 2 } = 57$

Find an equation in standard form for the ellipse that satisfies the given conditions. -The horizontal major axis is of length 18 , and the minor axis is of length 10 .

Match the given graph with its equation. -

Graph the parabola. - $( y - 8 ) ^ { 2 } = 6 ( x + 2 )$ $Graph the parabola. - ( y - 8 ) ^ { 2 } = 6 ( x + 2 )$

Find the standard form of the equation of the parabola. -Vertex at the origin, opens to the right, focal width $= 6$

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $x ^ { 2 } + x y + y ^ { 2 } + 4 x - 3 y - 3 = 0$

Find the standard form of the equation of the parabola. -Focus at $( 9 , - 5 )$ , directrix $y = - 15$

Functions and Graphs

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Trigonometric Functions

Analytic Trigonometry

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An Introduction to Calculus: Limits, Derivatives, and Integrals

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Exam 8: Analytic Geometry in Two and Three Dimensions

Graph the hyperbola. - 25y2−9x2=22525 y^{2}-9 x^{2}=22525y2−9x2=225

Determine the eccentricity, the type of conic, and the directrix. - r=79−6cos⁡θr = \frac { 7 } { 9 - 6 \cos \theta }r=9−6cosθ7​

Find an equation in standard form for the ellipse that satisfies the given conditions. -An ellipse with foci at (2,0)( 2,0 )(2,0) and (2,−6)( 2 , - 6 )(2,−6) ; major axis length of 10

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Center (4,−5)( 4 , - 5 )(4,−5) , focus (11,−5)( 11 , - 5 )(11,−5) , vertex (7,−5)( 7 , - 5 )(7,−5)

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - 9x2−6xy+6y2−20=09 x ^ { 2 } - 6 x y + 6 y ^ { 2 } - 20 = 09x2−6xy+6y2−20=0

Solve the problem. -The roof of a building is in the shape of the hyperbola y2−x2=11y ^ { 2 } - x ^ { 2 } = 11y2−x2=11 , where xxx and yyy are in meters. Refer to the figure and determine the height hhh of the outside walls. a=b=4ma=b=4 ma=b=4m

Find the vertex, focus, directrix, and focal width of the parabola. - x2=36y\mathrm { x } ^ { 2 } = 36 \mathrm { y }x2=36y

Find the vertex, the focus, and the directrix of the parabola. - y2+4x+8y+20=0y ^ { 2 } + 4 x + 8 y + 20 = 0y2+4x+8y+20=0

Find the center, vertices, and foci of the ellipse with the given equation. - x2225+y2625=1\frac { x ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 625 } = 1225x2​+625y2​=1

Solve the problem. -The roof of a building is in the shape of the hyperbola 8y2−x2=228 y ^ { 2 } - x ^ { 2 } = 228y2−x2=22 , where xxx and yyy are in meters. Refer to the figure and determine the height, h, of the outside walls. a=b=6ma = b = 6 ma=b=6m

Find the linear equations that can be used to convert an (x, y) equation to a (u, v) equation using the given angle of rotation - θ=80∘\theta = 80 ^ { \circ }θ=80∘

Solve the problem. -Find the vector from AAA to the midpoint of BCB CBC . A=(5,8,1),B=(2,7,6),C=(−9,7,4)\mathrm { A } = ( 5,8,1 ) , \mathrm { B } = ( 2,7,6 ) , \mathrm { C } = ( - 9,7,4 )A=(5,8,1),B=(2,7,6),C=(−9,7,4)

Use axis of rotation formulas for x and y to transform the quadratic equation to an equation in (u, v) coordinates with no cross-product term. - 7x2+63xy+13y2=577 x ^ { 2 } + 6 \sqrt { 3 } x y + 13 y ^ { 2 } = 577x2+63​xy+13y2=57