Question 1

Identify the graph of the ellipse. 9 x 2 + 4 y 2 $-$ 24 y + 11 = 0
A) 
B) 
C) 
D)

Accepted Answer

The answer of Identify the graph of the ellipse. 9...

Question 2

What is the graph of x ² + y ² = 36? A) point B) circle C) empty set

Accepted Answer

The given equation is in the form of a circle. We can rewrite it in standard form by completing the square:
(x S1U1P1)2 + (y S1U1P1)2 = 36/(2S1S1P0)2
This is the equation of a circle with center at the origin and radius 3. Therefore, the graph of the equation is a circle.

Question 3

Match each item with the correct graph to obtain a true statement.
-circle
A) $x ^ { 2 } + y ^ { 2 } = 0$ 
B) $x ^ { 2 } + y ^ { 2 } = 9$ 
C) $x ^ { 2 } + y ^ { 2 } = - 9$

Accepted Answer

A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The equation for a circle with center (0,0) and radius 3 is $x ^ { 2 } + y ^ { 2 } = 9$. The other two equations are not valid for circles because they either have a radius of 0 (A) or a negative radius, which does not make geometric sense (C).

Question 4

The hyperbola $- ( x + 3 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$ opens vertically.

Accepted Answer

The hyperbola has a negative sign in front of the x-term and a positive sign in front of the y-term, indicating that it opens vertically.

Question 5

Match each equation with the other equation that has the same graph.
-$3 y ^ { 2 } + 2 x ^ { 2 } = 25$ 
A) $3 x ^ { 2 } + 2 y ^ { 2 } = 25$ 
B) $2 x ^ { 2 } + 3 y ^ { 2 } = 25$ 
C) $x ^ { 2 } + y ^ { 2 } = 25$

Accepted Answer

The equation given is the equation of an ellipse with the center at the origin and major axis parallel to the y-axis. The equation B is also the equation of an ellipse with the same center and major axis parallel to the x-axis, thus having the same graph as the given equation. The equation A is also the equation of an ellipse, but with a different orientation and different intercepts, while equation C is the equation of a circle with the same center and radius as the given ellipse, thus having a different shape.

Question 6

The center of the hyperbola $( x - 1 ) ^ { 2 } - 9 ( x + 3 ) ^ { 2 } = 27$ is $( - 1,3 )$ .

Accepted Answer

The given equation is not correctly written to represent a hyperbola in standard form, and thus its center cannot be directly determined from the provided equation. Additionally, the notation seems to be incorrect or mistyped, as it mixes $x$ terms in a way that doesn't align with standard hyperbola equations.

Question 7

The graph of an equation of the form $A x ^ { 2 } + B y ^ { 2 } = C$ , where A , B , and C are nonzero real numbers and $A \neq B$ is an hyperbola.

Accepted Answer

The equation of an hyperbola in standard form is usually given as $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1$, where (h,k) is the center of the hyperbola and a and b are the length of the semi-axes. However, by rearranging the terms, we can obtain the equation $A x ^ { 2 } + B y ^ { 2 } = C$ , where A , B , and C are nonzero real numbers and $A \neq B$. This is the equation of a hyperbola with its center at the origin and its transverse axis along the x-axis or y-axis, depending on whether A or B is the larger coefficient. Therefore, the statement is true.

Question 8

What is the graph of x 2 + 2 y 2 = $-$ 25?
A) ellipse
B) empty set
C) point

Accepted Answer

The equation x^2 + 2y^2 = -25 cannot represent a real graph because the sum of squares cannot be negative. This means there are no real (x, y) solutions, hence the correct answer is an empty set.

Question 9

What is the graph of 5 x ² + 5 y ² = 0? A) circle B) point C) empty set

Accepted Answer

The given equation can be simplified as: 5⁽²)x + 5⁽²)y = 0 10(x+y) = 0 Since ^and are both nonzero constants, the only way the equation can be satisfied is if x+y=0. This means that the graph represents a line passing through the origin. Moreover, there is only one solution (0,0), which means that the graph is a point. Therefore, the answer is B) point.

Question 10

For all hyperbolas, the asymptotes intersect each other at the origin.

Accepted Answer

The asymptotes of a hyperbola intersect at the center of the hyperbola, which is not necessarily the origin unless the hyperbola is centered at the origin.

Question 11

Match each item with the correct graph to obtain a true statement.
-point
A) $x ^ { 2 } + 2 y ^ { 2 } = 0$ 
B) $x ^ { 2 } + 2 y ^ { 2 } = 16$ 
C) $x ^ { 2 } + 2 y ^ { 2 } = - 16$

Accepted Answer

All the points on graph A satisfy the equation $x ^ { 2 } + 2 y ^ { 2 } = 0$, which simplifies to $x^2 = -2y^2.$ The only way for this to be true is if both x and y are zero, which corresponds to the point (0,0).

Answer: B
 All the points on graph B satisfy the equation $x ^ { 2 } + 2 y ^ { 2 } = 16$, which represents an ellipse with x and y intercepts of $\pm4/\sqrt{3}$ and $\pm4/\sqrt{2}$, respectively.

Answer: None
 There are no real points that satisfy the equation $x^2 + 2y^2 = -16$, which is the equation for graph C. This is because $x^2$ and $2y^2$ are always nonnegative, so their sum can never be negative.

Question 12

Match each item with the correct graph to obtain a true statement.
-$\phi$ 
A) $x ^ { 2 } + 2 y ^ { 2 } = 0$ 
B) $x ^ { 2 } + 2 y ^ { 2 } = 16$ 
C) $x ^ { 2 } + 2 y ^ { 2 } = - 16$

Accepted Answer

The equation $x ^ { 2 } + 2 y ^ { 2 } = - 16$ has no real solutions, as adding non-negative values (squares of x and 2y) can never result in a negative number. Therefore, the graph for this equation does not exist. The other two graphs do exist and can be seen in the following images:

Graph A:

Graph B:

Neither of these graphs contain any real points where $x ^ { 2 } + 2 y ^ { 2 } = 0$, so neither match with $\phi$ (which could represent a solution to the equation). Therefore, the only possible match is with the non-existent graph C.

Question 13

Match each item with the correct graph to obtain a true statement.
-point
A) $x ^ { 2 } + y ^ { 2 } = 0$ 
B) $x ^ { 2 } + y ^ { 2 } = 9$ 
C) $x ^ { 2 } + y ^ { 2 } = - 9$

Accepted Answer

A) represents a single point at the origin (0,0) because the only solution to $x^2 + y^2 = 0$ is $x = 0$ and $y = 0$. B) represents a circle with a radius of 3 centered at the origin, and C) has no real solution because the sum of two squares cannot be negative.

Question 14

Match each equation with the other equation that has the same graph.
-$y ^ { 2 } + x ^ { 2 } = 25$ 
A) $3 x ^ { 2 } + 2 y ^ { 2 } = 25$ 
B) $2 x ^ { 2 } + 3 y ^ { 2 } = 25$ 
C) $x ^ { 2 } + y ^ { 2 } = 25$

Accepted Answer

All three equations represent ellipses, but only equation C has the same equation as the given equation. Equations A and B have different coefficients for the x and y terms, which changes the shape and orientation of the ellipse.

Question 15

Match each equation with the other equation that has the same graph.
-$2 y ^ { 2 } + 3 x ^ { 2 } = 25$ 
A) $3 x ^ { 2 } + 2 y ^ { 2 } = 25$ 
B) $2 x ^ { 2 } + 3 y ^ { 2 } = 25$ 
C) $x ^ { 2 } + y ^ { 2 } = 25$

Accepted Answer

The equations $2y^2 + 3x^2 = 25$ and $3x^2 + 2y^2 = 25$ are identical, just written in a different order, thus they represent the same graph.

Question 16

Match each item with the correct graph to obtain a true statement.
-$\phi$ 
A) $x ^ { 2 } + y ^ { 2 } = 0$ 
B) $x ^ { 2 } + y ^ { 2 } = 9$ 
C) $x ^ { 2 } + y ^ { 2 } = - 9$

Accepted Answer

The answer of Match each item with the correct graph...

Question 17

Identify the graph of the ellipse. x 2 + 2 x + 49 y 2 $-$ 80 = 0
A) 
B) 
C) 
D)

Accepted Answer

The answer of Identify the graph of the ellipse. x...

Question 18

Match each item with the correct graph to obtain a true statement.
-ellipse
A) $x ^ { 2 } + 2 y ^ { 2 } = 0$ 
B) $x ^ { 2 } + 2 y ^ { 2 } = 16$ 
C) $x ^ { 2 } + 2 y ^ { 2 } = - 16$

Accepted Answer

An ellipse is a closed curve that resembles a squashed or stretched circle. The equation of an ellipse is in the form of $\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1$, where (h,k) is the center of the ellipse, a is the distance from the center to the horizontal axis, and b is the distance from the center to the vertical axis. Option B is the only equation that fits this form, with the center at the origin, a=2, and b=4. Option A is the equation of a point, and option C is the equation of an imaginary shape.

Question 19

Is the graph of 16 x ² + 4 y ² = 9 the same as the graph of 4 x ² + 16 y ² = 9? Please enter yes if it is true or no on the contrary.

Accepted Answer

The answer of Is the graph of 16 x <sup>...

Question 20

What is the graph of 4 x ² + 16 y ² = 25? A) circle B) ellipse C) point

Accepted Answer

The given equation is in the standard form of an ellipse.

Quiz 12: Coordinate Geometry: Lines, Parabolas, Circles, Ellipses and Hyperbolas

Identify the graph of the ellipse. 9 x 2 + 4 y 2 $-$ 24 y + 11 = 0

What is the graph of x ² + y ² = 36?

Match each item with the correct graph to obtain a true statement. -circle

The hyperbola $- ( x + 3 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$ opens vertically.

Match each equation with the other equation that has the same graph. - $3 y ^ { 2 } + 2 x ^ { 2 } = 25$

The center of the hyperbola $( x - 1 ) ^ { 2 } - 9 ( x + 3 ) ^ { 2 } = 27$ is $( - 1,3 )$ .

The graph of an equation of the form $A x ^ { 2 } + B y ^ { 2 } = C$ , where A , B , and C are nonzero real numbers and $A \neq B$ is an hyperbola.

What is the graph of x 2 + 2 y 2 = $-$ 25?

What is the graph of 5 x ² + 5 y ² = 0?

For all hyperbolas, the asymptotes intersect each other at the origin.

Match each item with the correct graph to obtain a true statement. -point

Match each item with the correct graph to obtain a true statement. - $\phi$

Match each item with the correct graph to obtain a true statement. -point

Match each equation with the other equation that has the same graph. - $y ^ { 2 } + x ^ { 2 } = 25$

Match each equation with the other equation that has the same graph. - $2 y ^ { 2 } + 3 x ^ { 2 } = 25$

Match each item with the correct graph to obtain a true statement. - $\phi$

Identify the graph of the ellipse. x 2 + 2 x + 49 y 2 $-$ 80 = 0

Match each item with the correct graph to obtain a true statement. -ellipse

Is the graph of 16 x ² + 4 y ² = 9 the same as the graph of 4 x ² + 16 y ² = 9? Please enter yes if it is true or no on the contrary.

What is the graph of 4 x ² + 16 y ² = 25?

Quiz 12: Coordinate Geometry: Lines, Parabolas, Circles, Ellipses and Hyperbolas

Identify the graph of the ellipse. 9 x 2 + 4 y 2 −-− 24 y + 11 = 0

What is the graph of x 2 + y 2 = 36?

Match each item with the correct graph to obtain a true statement. -circle

The hyperbola −(x+3)2+(y−8)2=9- ( x + 3 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9−(x+3)2+(y−8)2=9 opens vertically.

Match each equation with the other equation that has the same graph. - 3y2+2x2=253 y ^ { 2 } + 2 x ^ { 2 } = 253y2+2x2=25

The center of the hyperbola (x−1)2−9(x+3)2=27( x - 1 ) ^ { 2 } - 9 ( x + 3 ) ^ { 2 } = 27(x−1)2−9(x+3)2=27 is (−1,3)( - 1,3 )(−1,3) .

The graph of an equation of the form Ax2+By2=CA x ^ { 2 } + B y ^ { 2 } = CAx2+By2=C , where A , B , and C are nonzero real numbers and A≠BA \neq BA=B is an hyperbola.

What is the graph of x 2 + 2 y 2 = −-− 25?

What is the graph of 5 x 2 + 5 y 2 = 0?

For all hyperbolas, the asymptotes intersect each other at the origin.

Match each item with the correct graph to obtain a true statement. -point

Match each item with the correct graph to obtain a true statement. - ϕ\phiϕ

Match each item with the correct graph to obtain a true statement. -point

Match each equation with the other equation that has the same graph. - y2+x2=25y ^ { 2 } + x ^ { 2 } = 25y2+x2=25

Match each equation with the other equation that has the same graph. - 2y2+3x2=252 y ^ { 2 } + 3 x ^ { 2 } = 252y2+3x2=25

Match each item with the correct graph to obtain a true statement. - ϕ\phiϕ

Identify the graph of the ellipse. x 2 + 2 x + 49 y 2 −-− 80 = 0

Match each item with the correct graph to obtain a true statement. -ellipse

Is the graph of 16 x 2 + 4 y 2 = 9 the same as the graph of 4 x 2 + 16 y 2 = 9? Please enter yes if it is true or no on the contrary.

What is the graph of 4 x 2 + 16 y 2 = 25?

Identify the graph of the ellipse. 9 x 2 + 4 y 2 $-$ 24 y + 11 = 0

What is the graph of x ² + y ² = 36?

The hyperbola $- ( x + 3 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$ opens vertically.

Match each equation with the other equation that has the same graph. - $3 y ^ { 2 } + 2 x ^ { 2 } = 25$

The center of the hyperbola $( x - 1 ) ^ { 2 } - 9 ( x + 3 ) ^ { 2 } = 27$ is $( - 1,3 )$ .

The graph of an equation of the form $A x ^ { 2 } + B y ^ { 2 } = C$ , where A , B , and C are nonzero real numbers and $A \neq B$ is an hyperbola.

What is the graph of x 2 + 2 y 2 = $-$ 25?

What is the graph of 5 x ² + 5 y ² = 0?

Match each item with the correct graph to obtain a true statement. - $\phi$

Match each equation with the other equation that has the same graph. - $y ^ { 2 } + x ^ { 2 } = 25$

Match each equation with the other equation that has the same graph. - $2 y ^ { 2 } + 3 x ^ { 2 } = 25$

Match each item with the correct graph to obtain a true statement. - $\phi$

Identify the graph of the ellipse. x 2 + 2 x + 49 y 2 $-$ 80 = 0

Is the graph of 16 x ² + 4 y ² = 9 the same as the graph of 4 x ² + 16 y ² = 9? Please enter yes if it is true or no on the contrary.

What is the graph of 4 x ² + 16 y ² = 25?