Question 1

Which of the following equations represents a hyperbola centered at the origin?

Accepted Answer

$$x^2 = 25$$ + $$5y^2$$

Question 2

Given the equation $$x^2 - 6x + 8y - 7 = 0$, what type of conic section does it represent and what is its vertex?

Accepted Answer

Parabola, vertex at $$(3, 2)$$

Question 3

Which condition must be true for the general second-degree equation $$Ax^2 + Dx$$ + $$Cy^2 + Ey + F = 0$ to represent a circle?

Accepted Answer

$$A = C 
eq 0$$

Question 4

Find the eccentricity of the ellipse given by $$\frac{x^2$}{9} + \frac{y^2$}{16} = 1.$$

Accepted Answer

$$e = \frac{\sqrt{7}}{4}$$

Question 5

A planet orbits the sun in an ellipse with eccentricity $$e = 0.0935$$ and its closest distance to the sun is $$128.5$$ million miles. What is the maximum distance from the sun?

Accepted Answer

$$155.1$$ million miles

Question 6

Which of the following best describes the eccentricity $$e of a $$hyperbola?

Accepted Answer

$$e \u003e 1$$

Question 7

Given the system $$x^2$$ + $$y^2 = 4$ and 2x^2$ - y^2$ = 8$$, what are the solutions?

Accepted Answer

$$(2, 0)$$ and $$(-2, 0)$$

Question 8

Which of the following equations represents an ellipse centered at $$(2, -3)$$?

Accepted Answer

$$\frac{(x-2)^2$}{9} + \frac{(y+3)^2$}{4} = 1$$

Question 9

If a conic section has the equation $$4y^2 - 16y$$ - $$9x^2 + 18x = -43$, what is its center?

Accepted Answer

$$(1, 2)$$

Question 10

Which of the following is the correct formula for the eccentricity $$e of an $$ellipse with $$a \u003e b$$?

Accepted Answer

$$e = \frac{c}{a}$$, where $$c = \sqrt{a^2$ - b^2$}$$

Question 11

A conic section has a focus at $$(3, 0)$$ and eccentricity $$e = 2. $$What is its equation?

Accepted Answer

$$\frac{x^2$}{9/4} - \frac{y^2$}{27/4} = 1$$

Question 12

Which of the following best describes the region that satisfies both $$\frac{x^2$}{25} + \frac{y^2$}{9} \leq 1$$ and $$\frac{x^2$}{4} - \frac{y^2$}{9} \geq 1$$?

Accepted Answer

The intersection of the interior of an ellipse and the exterior of a hyperbola

Question 13

Given the system $$x^2 + 3xy$$ + $$y^2 = 22$ and x^2$ - xy + y^2$ = 6$$, what is the value of $$y in $$terms of $$x$$ after elimination?

Accepted Answer

$$y = \frac{4}{x}$$

Question 14

Which of the following statements about the eccentricity of ellipses is true?

Accepted Answer

Ellipses with eccentricity close to 0 are nearly circular.

Question 15

If the equation of a conic is $$x^2$$ - $$y^2 = 1$, what type of conic is it?

Accepted Answer

Hyperbola

Question 16

Given the ellipse $$\frac{y^2$}{64} + \frac{x^2$}{48} = 1$$, what is the length of the major axis?

Accepted Answer

$$8$$

Question 17

Which of the following is a real-world application of ellipses discussed in the materials?

Accepted Answer

The orbit of a planet around the sun

Question 18

If $$e = \frac{1}{2}$$ and the vertex is at $$(0, -8)$$, what is the equation of the ellipse?

Accepted Answer

$$\frac{y^2$}{64} + \frac{x^2$}{48} = 1$$

Conic Sections and Nonlinear Systems: Study Notes

Guida di studio - Domande di pratica

Study guide - Flashcard