Understanding Ellipses

The video tutorial introduces ellipses, defining them as a set of points where the sum of distances from two fixed points is constant. It explains the characteristics of ellipses, including major and minor axes, vertices, and foci. The tutorial discusses the variables a, b, and c, and how they relate to the ellipse's equation. It also provides a step-by-step guide on graphing an ellipse centered at the origin, using the Pythagorean theorem to find the foci.

What is the defining characteristic of an ellipse?

• The ratio of distances from two fixed points is constant.

• The product of distances from two fixed points is constant.

• The difference of distances from two fixed points is constant.

• The sum of distances from two fixed points is constant.

Which axis is longer in an ellipse?

• Semi-minor axis

• Minor axis

• Major axis

• Semi-major axis

What are the points at the ends of the major axis called?

• Co-vertices

• Foci

• Centers

• Vertices

In the context of an ellipse, what does the variable 'a' represent?

• Distance from the center to a co-vertex

• Distance from the center to the origin

• Distance from the center to a vertex

• Distance from the center to a focus

Which variable is not directly included in the standard form equation of an ellipse?

• a

• b

• c

• x

What is the standard form equation of an ellipse centered at the origin?

• x^2 + y^2 = 1

• x^2 - y^2 = 1

• x^2/a^2 + y^2/b^2 = 1

• x^2/a + y^2/b = 1

How do you calculate the length of the major axis?

• 2 times the length of b

• 2 times the length of c

• 2 times the length of a

• 2 times the length of the semi-minor axis

What theorem is used to find the foci of an ellipse?

• Remainder theorem

• Binomial theorem

• Fundamental theorem of calculus

• Pythagorean theorem

If a^2 = 9 and b^2 = 4, what is the value of c^2?

• 7

• 5

• 13

• 3

What are the coordinates of the foci for an ellipse centered at the origin with a = 3 and b = 2?

• (±2, 0)

• (±√5, 0)

• (±3, 0)

• (0, ±2)