Question 1

Identify the type of graph.
-$6 x ^ { 2 } - 12 x + 3 y ^ { 2 } + 18 y - 21 = 0$

Accepted Answer

The given equation is of the form $Ax^2 + Cy^2 + Dx + Ey + F = 0$, where $A$ and $C$ are both positive and not equal, indicating an ellipse. Completing the square for both $x$ and $y$ terms and dividing the whole equation by the coefficient of $x^2$ and $y^2$ (which are equal in this case) would give an equation in the standard form of an ellipse. The center of the ellipse can be found by transforming the equation into its standard form, which reveals the center to be at (1, -3).

Question 2

Identify the type of conic section.
-Identify the type of conic section consisting of the set of all points in the plane for which the distance from the point (-2, 0) is one-fourth of the distance from the line x = -9.

Accepted Answer

This is an ellipse because the set of points where the distance to a point (focus) is a constant fraction (less than 1) of the distance to a line (directrix) forms an ellipse.

Question 3

Graph the conic section.
-$$x ^ { 2 } - 14 x - 4 y + 73 = 0$$

Accepted Answer

The answer of Graph the conic section.
-\[x ^ { 2...

Question 4

Identify the type of graph.
-$x ^ { 2 } - 9 x + y = 0$

Accepted Answer

The answer of Identify the type of graph.
-\(x ^ {...

Question 5

Identify the type of conic section.
-Identify the type of conic section consisting of the set of all points in the plane for which the sum of the distances from the points (4, 0) and (-4, 0) is 28.

Accepted Answer

The given set of points describes an ellipse, as an ellipse is defined by the set of points where the sum of the distances from two fixed points (the foci) is constant. In this case, the two fixed points are (4, 0) and (-4, 0), and the constant sum of distances is 28.

Question 6

Graph the conic section.
-$2 x^{2}+4 y^{2}-28 x+32 y+154=0$

Accepted Answer

The answer of Graph the conic section.
-\(2 x^{2}+4 y^{2}-28 x+32...

Question 7

Identify the type of graph.
-$\frac { x ^ { 2 } } { 12 } = 1 - \frac { y ^ { 2 } } { 3 }$

Accepted Answer

The answer of Identify the type of graph.
-\(\frac { x...

Question 8

Identify the type of conic section.
-Identify the type of conic section consisting of the set of all points in the plane for which the distance from the point (0, 19) is twice the distance from the line y = 5.

Accepted Answer

This conic section is a hyperbola because the set of points for which the distance to a point (focus) is a constant multiple (in this case, twice) of the distance to a line (directrix) forms a hyperbola.

Question 9

Identify the type of graph.
-$\frac { ( x - 7 ) ^ { 2 } } { 3 } + \frac { ( y - 3 ) ^ { 2 } } { 7 } = 0$

Accepted Answer

The equation represents a single point because both the numerator and the denominator of each fraction are squared, and the equation equals 0. This means that both $x - 7$ and $y - 3$ must be 0, so the graph is a single point at (7, 3).

Question 10

Identify the type of graph.
-$$- x ^ { 2 } - 10 x + 6 y ^ { 2 } + 12 y = 16$$

Accepted Answer

The equation is a hyperbola because it has both $x^2$ and $y^2$ terms with opposite signs.

Question 11

Identify the type of graph.
-$10 x ^ { 2 } + 60 x + 10 y ^ { 2 } - 20 y = - 99$

Accepted Answer

The given equation can be rewritten in the standard form of a circle's equation by completing the squares for both x and y terms. Rearranging and completing the squares gives us $(x+3)^2 + (y-1)^2 = 100$, which represents a circle with center (-3, 1).

Question 12

Graph the conic section.
-$$x ^ { 2 } - 9 y - 6 x = 0$$

Accepted Answer

The answer of Graph the conic section.
-\[x ^ { 2...

Question 13

Identify the type of graph.
-$( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + 11 = 0$

Accepted Answer

The equation given is of the form $(x - h)^2 + (y - k)^2 = r^2$, which typically represents a circle with center $(h, k)$ and radius $r$. However, in this case, the right side of the equation is negative ($11 = 0$), which is impossible for a circle's equation since the radius squared ($r^2$) cannot be negative. Therefore, this equation does not represent a graphable figure in real number coordinates.

Question 14

Graph the conic section.
-$$x ^ { 2 } - 14 x - 8 y + 81 = 0$$

Accepted Answer

The answer of Graph the conic section.
-\[x ^ { 2...

Question 15

Identify the equation as a parabola, circle, ellipse, or hyperbola.
-$( x - 5 ) ^ { 2 } = 36 - y ^ { 2 }$

Accepted Answer

The answer of Identify the equation as a parabola, circle,...

Question 16

Identify the type of graph.
-$$- 3 x ^ { 2 } + 24 x + 2 y ^ { 2 } - 24 y - 48 = - 72$$

Accepted Answer

The answer of Identify the type of graph.
-\[- 3 x...

Question 17

Graph the conic section.
-$$y ^ { 2 } - 12 y + 4 x + 48 = 0$$

Accepted Answer

The answer of Graph the conic section.
-\[y ^ { 2...

Question 18

Identify the type of conic section.
-Identify the type of conic section consisting of the set of all points in the plane for which the absolute value of the difference of the distances from the points (0, 7) and (0, -7) is 3.

Accepted Answer

The given set of points describes a hyperbola, as it involves the absolute value of the difference of distances from two fixed points (the foci), which is a defining characteristic of a hyperbola.

Question 19

Identify the type of graph.
-$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 36 } = 1$

Accepted Answer

The equation is in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a^2 = b^2 = 36$, indicating it is a circle (since $a = b$) with a radius of 6 and centered at the origin (0, 0).

Question 20

Identify the type of graph.
-$\frac { x ^ { 2 } } { 11 } = 1 + \frac { y ^ { 2 } } { 5 }$

Accepted Answer

The equation can be rearranged to the standard form of a hyperbola, $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, indicating it is a hyperbola that opens to the right and left.

Quiz 11: Analytic Geometry

Identify the type of graph. - $6 x ^ { 2 } - 12 x + 3 y ^ { 2 } + 18 y - 21 = 0$

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the distance from the point (-2, 0) is one-fourth of the distance from the line x = -9.

Graph the conic section. - $x ^ { 2 } - 14 x - 4 y + 73 = 0$ $Graph the conic section. - x ^ { 2 } - 14 x - 4 y + 73 = 0 A) B) C) D)$

Identify the type of graph. - $x ^ { 2 } - 9 x + y = 0$

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the sum of the distances from the points (4, 0) and (-4, 0) is 28.

Graph the conic section. - $2 x^{2}+4 y^{2}-28 x+32 y+154=0$ $Graph the conic section. - 2 x^{2}+4 y^{2}-28 x+32 y+154=0 A) B) C) D)$

Identify the type of graph. - $\frac { x ^ { 2 } } { 12 } = 1 - \frac { y ^ { 2 } } { 3 }$

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the distance from the point (0, 19) is twice the distance from the line y = 5.

Identify the type of graph. - $\frac { ( x - 7 ) ^ { 2 } } { 3 } + \frac { ( y - 3 ) ^ { 2 } } { 7 } = 0$

Identify the type of graph. - $- x ^ { 2 } - 10 x + 6 y ^ { 2 } + 12 y = 16$

Identify the type of graph. - $10 x ^ { 2 } + 60 x + 10 y ^ { 2 } - 20 y = - 99$

Graph the conic section. - $x ^ { 2 } - 9 y - 6 x = 0$ $Graph the conic section. - x ^ { 2 } - 9 y - 6 x = 0 A) B) C) D)$

Identify the type of graph. - $( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + 11 = 0$

Graph the conic section. - $x ^ { 2 } - 14 x - 8 y + 81 = 0$ $Graph the conic section. - x ^ { 2 } - 14 x - 8 y + 81 = 0 A) B) C) D)$

Identify the equation as a parabola, circle, ellipse, or hyperbola. - $( x - 5 ) ^ { 2 } = 36 - y ^ { 2 }$

Identify the type of graph. - $- 3 x ^ { 2 } + 24 x + 2 y ^ { 2 } - 24 y - 48 = - 72$

Graph the conic section. - $y ^ { 2 } - 12 y + 4 x + 48 = 0$ $Graph the conic section. - y ^ { 2 } - 12 y + 4 x + 48 = 0 A) B) C) D)$

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the absolute value of the difference of the distances from the points (0, 7) and (0, -7) is 3.

Identify the type of graph. - $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 36 } = 1$

Identify the type of graph. - $\frac { x ^ { 2 } } { 11 } = 1 + \frac { y ^ { 2 } } { 5 }$

Quiz 11: Analytic Geometry

Identify the type of graph. - 6x2−12x+3y2+18y−21=06 x ^ { 2 } - 12 x + 3 y ^ { 2 } + 18 y - 21 = 06x2−12x+3y2+18y−21=0

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the distance from the point (-2, 0) is one-fourth of the distance from the line x = -9.

Graph the conic section. - x2−14x−4y+73=0x ^ { 2 } - 14 x - 4 y + 73 = 0x2−14x−4y+73=0

Identify the type of graph. - x2−9x+y=0x ^ { 2 } - 9 x + y = 0x2−9x+y=0

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the sum of the distances from the points (4, 0) and (-4, 0) is 28.

Graph the conic section. - 2x2+4y2−28x+32y+154=02 x^{2}+4 y^{2}-28 x+32 y+154=02x2+4y2−28x+32y+154=0

Identify the type of graph. - x212=1−y23\frac { x ^ { 2 } } { 12 } = 1 - \frac { y ^ { 2 } } { 3 }12x2​=1−3y2​

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the distance from the point (0, 19) is twice the distance from the line y = 5.

Identify the type of graph. - (x−7) 23+(y−3) 27=0\frac { ( x - 7 ) ^ { 2 } } { 3 } + \frac { ( y - 3 ) ^ { 2 } } { 7 } = 03(x−7) 2​+7(y−3) 2​=0

Identify the type of graph. - −x2−10x+6y2+12y=16- x ^ { 2 } - 10 x + 6 y ^ { 2 } + 12 y = 16−x2−10x+6y2+12y=16

Identify the type of graph. - 10x2+60x+10y2−20y=−9910 x ^ { 2 } + 60 x + 10 y ^ { 2 } - 20 y = - 9910x2+60x+10y2−20y=−99

Graph the conic section. - x2−9y−6x=0x ^ { 2 } - 9 y - 6 x = 0x2−9y−6x=0

Identify the type of graph. - (x+6) 2+(y−3) 2+11=0( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + 11 = 0(x+6) 2+(y−3) 2+11=0

Graph the conic section. - x2−14x−8y+81=0x ^ { 2 } - 14 x - 8 y + 81 = 0x2−14x−8y+81=0

Identify the equation as a parabola, circle, ellipse, or hyperbola. - (x−5) 2=36−y2( x - 5 ) ^ { 2 } = 36 - y ^ { 2 }(x−5) 2=36−y2

Identify the type of graph. - −3x2+24x+2y2−24y−48=−72- 3 x ^ { 2 } + 24 x + 2 y ^ { 2 } - 24 y - 48 = - 72−3x2+24x+2y2−24y−48=−72

Graph the conic section. - y2−12y+4x+48=0y ^ { 2 } - 12 y + 4 x + 48 = 0y2−12y+4x+48=0

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the absolute value of the difference of the distances from the points (0, 7) and (0, -7) is 3.

Identify the type of graph. - x236+y236=1\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 36 } = 136x2​+36y2​=1

Identify the type of graph. - x211=1+y25\frac { x ^ { 2 } } { 11 } = 1 + \frac { y ^ { 2 } } { 5 }11x2​=1+5y2​

Identify the type of graph. - $6 x ^ { 2 } - 12 x + 3 y ^ { 2 } + 18 y - 21 = 0$

Graph the conic section. - $x ^ { 2 } - 14 x - 4 y + 73 = 0$ $Graph the conic section. - x ^ { 2 } - 14 x - 4 y + 73 = 0 A) B) C) D)$

Identify the type of graph. - $x ^ { 2 } - 9 x + y = 0$

Graph the conic section. - $2 x^{2}+4 y^{2}-28 x+32 y+154=0$ $Graph the conic section. - 2 x^{2}+4 y^{2}-28 x+32 y+154=0 A) B) C) D)$

Identify the type of graph. - $\frac { x ^ { 2 } } { 12 } = 1 - \frac { y ^ { 2 } } { 3 }$

Identify the type of graph. - $\frac { ( x - 7 ) ^ { 2 } } { 3 } + \frac { ( y - 3 ) ^ { 2 } } { 7 } = 0$

Identify the type of graph. - $- x ^ { 2 } - 10 x + 6 y ^ { 2 } + 12 y = 16$

Identify the type of graph. - $10 x ^ { 2 } + 60 x + 10 y ^ { 2 } - 20 y = - 99$

Graph the conic section. - $x ^ { 2 } - 9 y - 6 x = 0$ $Graph the conic section. - x ^ { 2 } - 9 y - 6 x = 0 A) B) C) D)$

Identify the type of graph. - $( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + 11 = 0$

Graph the conic section. - $x ^ { 2 } - 14 x - 8 y + 81 = 0$ $Graph the conic section. - x ^ { 2 } - 14 x - 8 y + 81 = 0 A) B) C) D)$

Identify the equation as a parabola, circle, ellipse, or hyperbola. - $( x - 5 ) ^ { 2 } = 36 - y ^ { 2 }$

Identify the type of graph. - $- 3 x ^ { 2 } + 24 x + 2 y ^ { 2 } - 24 y - 48 = - 72$

Graph the conic section. - $y ^ { 2 } - 12 y + 4 x + 48 = 0$ $Graph the conic section. - y ^ { 2 } - 12 y + 4 x + 48 = 0 A) B) C) D)$

Identify the type of graph. - $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 36 } = 1$

Identify the type of graph. - $\frac { x ^ { 2 } } { 11 } = 1 + \frac { y ^ { 2 } } { 5 }$