Algebra and Trigonometry: Analytic Geometry flashcards

Question 1

What is a conic section?

Accepted Answer

A shape made where a plane cuts a right circular cone

Question 2

Which four shapes are the nondegenerate conic sections?

Accepted Answer

Circles, ellipses, hyperbolas and parabolas (the plane does not pass through the cone's apex)

Question 3

What shapes can a degenerate conic section be?

Accepted Answer

A point, a line, or intersecting lines (the plane passes through the apex of a double cone)

Question 4

What is an ellipse, defined by distances?

Accepted Answer

All points in a plane whose distances to two fixed points add up to a constant

Question 5

What is a hyperbola, defined by distances?

Accepted Answer

All points in a plane whose distances to the two foci differ by a positive constant

Question 6

What is a parabola, defined by distances?

Accepted Answer

All points in a plane equally far from a fixed line (directrix) and a fixed point (focus) not on that line

Question 7

What is the major axis of an ellipse?

Accepted Answer

The longer of its two axes

Question 8

What is the minor axis of an ellipse?

Accepted Answer

The shorter of its two axes

Question 9

Where is the center of an ellipse?

Accepted Answer

At the midpoint of both its major and minor axes

Question 10

Where is the center of a hyperbola?

Accepted Answer

At the midpoint of both its transverse and conjugate axes

Question 11

Which axis of a hyperbola contains the foci and ends at the vertices?

Accepted Answer

The transverse axis

Question 12

Which axis of a hyperbola is perpendicular to the transverse axis and ends at the co-vertices?

Accepted Answer

The conjugate axis

Question 13

What is the latus rectum of a parabola?

Accepted Answer

The segment through the focus, parallel to the directrix, with both endpoints on the parabola

Question 14

On which line does the focus of a parabola lie?

Accepted Answer

On its axis of symmetry

Question 15

How is eccentricity e defined for a point P on a conic with focus F and directrix D?

Accepted Answer

e = PF / PD, a positive real number

Question 16

Standard form of a horizontal ellipse centered at the origin?

Accepted Answer

x^2/a^2 + y^2/b^2 = 1, with a > b

Question 17

Standard form of a hyperbola centered at the origin with transverse axis on the x-axis?

Accepted Answer

x^2/a^2 − y^2/b^2 = 1

Question 18

Standard form of a parabola with vertex at the origin and the y-axis as axis of symmetry?

Accepted Answer

x^2 = 4py

Question 19

For y^2 = 4px, which way does the parabola open when p > 0, and when p < 0?

Accepted Answer

Right when p > 0; left when p < 0

Question 20

For x^2 = 4py, which way does the parabola open when p > 0, and when p < 0?

Accepted Answer

Up when p > 0; down when p < 0

Question 21

What does an xy term in a conic's equation tell you?

Accepted Answer

The conic has been rotated about the origin

Question 22

Under a rotation of axes by θ, what is x in terms of x′ and y′?

Accepted Answer

x = x′ cos θ − y′ sin θ

Question 23

Under a rotation of axes by θ, what is y in terms of x′ and y′?

Accepted Answer

y = x′ sin θ + y′ cos θ

Question 24

Which formula gives the angle of rotation θ for Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0?

Accepted Answer

cot(2θ) = (A − C)/B

Question 25

Which polar-coordinate identities give x and y in terms of r and θ?

Accepted Answer

x = r cos θ and y = r sin θ

Algebra and Trigonometry: Analytic Geometry flashcards

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