Conics and Parametrics Review

Flashcards for reviewing conic sections, parametric equations, polar coordinates, and projectile motion.

What is the standard form equation for a vertical parabola?

(x-h)^2 = 4p(y-k)

For a vertical parabola, what are the coordinates of the vertex?

(h,k)

For a vertical parabola, what are the coordinates of the focus?

(h,k+p)

What is the equation for the axis of symmetry for a vertical parabola?

x=h

What is the equation for the directrix of a vertical parabola?

y = k-p

What is the standard form equation for a horizontal parabola?

(y-k)^2 = 4p(x-h)

For a horizontal parabola, what are the coordinates of the focus?

(h+p, k)

What is the equation for the axis of symmetry for a horizontal parabola?

y=k

What is the equation for the directrix of a horizontal parabola?

x = h-p

What is the general form for a vertical ellipse?

(x-h)^2/b^2 + (y-k)^2/a^2 = 1

For a vertical ellipse, what are the coordinates of the center?

(h,k)

For a vertical ellipse, what are the coordinates of the foci?

(h,k+ or - c)

For a vertical ellipse, what are the coordinates of the vertices?

(h, k+ or - a)

For a vertical ellipse, what are the coordinates of the co-vertices?

(h+ or - b, k)

What is the equation of the major axis for a vertical ellipse?

x=h

What is the equation of the minor axis for a vertical ellipse?

y=k

What is the relationship between a, b, and c in an ellipse?

C^2=a^2-b^2

What is the general form for a horizontal ellipse?

(x-h)^2/a^2 + (y-k)^2/b^2 = 1

For a horizontal ellipse, what are the coordinates of the foci?

(h+ or - c,k)

For a horizontal ellipse, what are the coordinates of the vertices?

(h + or - a, k)

For a horizontal ellipse, what are the coordinates of the co-vertices?

(h, k+ or - b)

What is the equation of the minor axis for a horizontal ellipse?

x=h

What is the equation of the major axis for a horizontal ellipse?

y=k

What is the general form for a circle?

(x-h)^2 + (y-k)^2 = r^2

For a circle, what are the coordinates of the center?

(h,k)

What does 'R' represent in the equation of a circle?

radius

What is the general form for a horizontal hyperbola?

(x-h)^2/a^2 - (y-k)^2/b^2 = 1

For a horizontal hyperbola, what are the coordinates of the center?

(h,k)

For a horizontal hyperbola, what are the coordinates of the vertices?

(h + or - a, k)

For a horizontal hyperbola, what are the coordinates of the foci?

(h+ or - c, k)

What is the equation of the transverse axis for a horizontal hyperbola?

y=k

What is the equation of the conjugate axis for a horizontal hyperbola?

x=h

What is the equation for the asymptotes of a horizontal hyperbola?

y-k = + or - b/a * (x-h)

What is the relationship between a, b, and c in a hyperbola?

c^2=a^2+b^2

What is the general form for a vertical hyperbola?

(y-k)^2/a^2 - (x-h)^2/b^2 = 1

For a vertical hyperbola, what are the coordinates of the vertices?

(h, k + or - a)

For a vertical hyperbola, what are the coordinates of the foci?

(h, k+ or - c)

What is the equation of the transverse axis for a vertical hyperbola?

x=h

What is the equation of the conjugate axis for a vertical hyperbola?

y=k

What is the equation for the asymptotes of a vertical hyperbola?

y-k = + or - a/b * (x-h)

What is the general form of a conic equation that we use the discriminant with?

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

What is the formula for the discriminant?

b^2 - 4ac

If b^2 - 4ac < 0 and b = 0 and A=C, what type of conic is it?

Circle

If b^2 - 4ac < 0 and b ≠ 0 and A≠C, what type of conic is it?

Ellipse

If b^2 - 4ac = 0, what type of conic is it?

Parabola

If b^2 - 4ac > 0, what type of conic is it?

Hyperbola

If e = 0, what type of conic is it?

Circle

If e = 1, what type of conic is it?

Parabola

If e < 1, what type of conic is it?

Ellipse

If e > 1, what type of conic is it?

Hyperbola

How do you calculate speed between two points using parametrics?

t = d/s (Distance / Speed)

How do you calculate the slope to parametrize a line between two points?

x - x1 = slope(t - t1), y - y1 = slope(t - t1)

What does 'e' represent in polar coordinates and conics?

eccentricity

What is the formula for eccentricity?

e = c/a

What does 'd' represent in polar coordinates and conics?

distance from pole/origin

What is the polar equation if the directrix is perpendicular to the polar axis at a distance d units to the left of the pole?

r = ed/ 1 + ecosθ

What is the polar equation if the directrix is perpendicular to the polar axis at a distance d units to the right of the pole?

r = ed/ 1 - ecosθ

What is the polar equation if the directrix is parallel to the polar axis at a distance d units above the pole?

r = ed/ 1 + esinθ

What is the polar equation if the directrix is parallel to the polar axis at a distance d units below the pole?

r = ed/ 1 - esinθ

What is the formula for horizontal distance in projectile motion?

x = tvcosθ

What is the formula for vertical position in projectile motion?

y = tvsinθ - (gt^2)/2 + h

What are the parametric equations for a Ferris wheel when using CCW motion and starting in quadrant I?

X = h - rsinθ, y = k + rcosθ

What are the parametric equations for a Ferris wheel when using CW motion and starting in quadrant I?

X = h + rsinθ, y = k - rcosθ