Chapter 10 (conics)

Question 1

What is the formula for a circle and how would you find center and radius?

Answer 1

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

(h,k) is center (if none center at (0,0)
R is radius
From center to point on circle, radius is square root of (x-h)^2 + (y-k)^2

Question 2

What is the definition of a ellipse? (Set of all points and from a double cone)

Answer 2

The set of all points in a plane, the sum of whose distances from two fixed points (foci) is a constant

Tilted plane neither parallel nor perpendicular to the axis going through the centers of a double cone and only going through one cone

Question 3

What is the formula for a eclipse and how would you find the center, foci, and vertices?

Answer 3

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

(h,k) is center (if none center is at (0,0)
A is distance from center on x-axis (vertices)
B is distance from center on y-axis (vertices)
C^2 = |a^2 - b^2| where c is the distance from the center to the foci on the major axis (longest one)

Question 4

What is the definition of a hyperbola? (Set of all points and from a double cone)

Answer 4

The set of all points in a plane the difference of whose distances from two fixed points (foci) is a positive constant

Plane perpendicular to the axis going through the centers of a double cone (but not going through the center)

Question 5

What are the formulas for a hyperbola and how would you find the center, vertices, foci and asymptotes?

Answer 5

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

Center is at (h,k) (if none center at (0,0))
Asymptotes Y-k=+/- b/a (x-k)
A is distance from center on x-axis (vertices)
B is distance from center on y-axis (vertices)
C^2 = a^2 + b^2 where c is the distance from the center to the foci

Question 6

What is the definition of a circle? (Set of all points and from a double cone)

Answer 6

The set of all points in a plane equidistant from a given point (center)

Plane parallel to the axis going through the centers of a double cone

Question 7

What is the definition of a parabola? (Set of all points and double cone)

Answer 7

The set of all points in a plane equidistant from a given point (focus) and a given line (directrix)

Question 8

What are the formulas for a parabola and how would you find vertex, focus and directrix?

Answer 8

Open up- Y=(1/4p)X^2
Open down- Y=-(1/4p)X^2
Open left- X=-(1/4p)Y^2
Open right- X=(1/4p)Y^2

X and Y will be minus vertex, p is distance to focus and directrix