Understanding Hyperbolas and Eccentricity

The video tutorial explains how to determine the eccentricity of a hyperbola given a tangent line and its intersection with a directrix. It begins by setting up the problem and drawing the hyperbola, explaining the concepts of foci and directrices. The tutorial then uses algebraic manipulation and known formulas to solve for the eccentricity, concluding that the eccentricity is 2.

What is the general form of a hyperbola that opens left and right?

• x^2/a^2 - y^2/b^2 = 1

• y^2/a^2 - x^2/b^2 = 1

• x^2 - y^2 = 1

• x^2 + y^2 = 1

Where does the line 2x + y = 1 intersect the x-axis?

• (1/2, 0)

• (0, 1)

• (0, 1/2)

• (1, 0)

What is the relationship between the eccentricity and the directrix of a hyperbola?

• Eccentricity is the sum of the distances to the foci

• Eccentricity is the product of the distances to the foci

• Eccentricity is the ratio of the distance to the focus over the distance to the directrix

• Eccentricity is the distance to the directrix

How is the focal length of a hyperbola related to its parameters a and b?

• Focal length is a^2 / b^2

• Focal length is a^2 * b^2

• Focal length is a^2 + b^2

• Focal length is a^2 - b^2

What is the formula for the directrix distance in terms of eccentricity and a?

• a * eccentricity

• a / eccentricity

• eccentricity / a

• a + eccentricity

What is the value of the eccentricity if a hyperbola's directrix is at x = 1/2?

• 3

• 1

• 2

• 1/2

Which equation represents the relationship between c, b, m, and a for a tangent line to a hyperbola?

• c^2 + b^2 = m^2 * a^2

• c^2 - b^2 = m^2 * a^2

• c^2 * b^2 = m^2 + a^2

• c^2 / b^2 = m^2 / a^2

What is the result of solving the polynomial equation for eccentricity?

• Eccentricity is -1

• Eccentricity is 2

• Eccentricity is 1

• Eccentricity is 0

Why is the eccentricity of a hyperbola always greater than 1?

• Because it is defined as such

• Because it is the ratio of distances

• Because it is a property of parabolas

• Because it is a property of ellipses

What is the correct factorization of the polynomial equation derived in the solution?

• (epsilon^2 + 2)(epsilon^2 - 2)

• (epsilon^2 - 2)(epsilon^2 - 2)

• (epsilon^2 + 4)(epsilon^2 + 1)

• (epsilon^2 - 4)(epsilon^2 - 1)