What is the standard equation of a hyperbola centered at the origin with its transverse axis along the x-axis?

The standard equation is x²/a² - y²/b² = 1, where a is the semi-major axis and b is the semi-minor axis.

If the foci of a hyperbola are at (±5,0) and the length of the transverse axis is 6, what is the equation of the hyperbola?

Here, 2a = 6, so a = 3. The focal distance c = 5. Using b² = c² - a², we get b² = 5² - 3² = 25 - 9 = 16. Thus, the equation is x²/9 - y²/16 = 1.

True/False: The equation of a hyperbola with a vertical transverse axis is given by x²/a² - y²/b² = 1.

False. The correct equation is y²/a² - x²/b² = 1 when the transverse axis is along the y-axis.

Fill in the blank: The distance between the foci of a hyperbola is equal to ___ times the length of the transverse axis.

The distance between the foci of a hyperbola is equal to 2c, which is greater than the length of the transverse axis.

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

For a hyperbola, the relationship is given by c² = a² + b², where c is the distance from the center to each focus, a is the semi-major axis, and b is the semi-minor axis.

Define the term 'latus rectum' of a hyperbola and provide its length.

The latus rectum of a hyperbola is a line segment perpendicular to the transverse axis and passing through a focus. Its length is given by 2b²/a.

Fill in the blank: The equation of the auxiliary circle for a hyperbola with transverse axis length 2a is ___

The equation of the auxiliary circle is x² + y² = a².

What is the parametric form of a hyperbola?

The parametric equations of a hyperbola are x = a sec θ and y = b tan θ, where θ is the parameter.

True/False: The hyperbola x² - y² = a² is called a rectangular hyperbola.

True. It is called a rectangular hyperbola because its asymptotes are perpendicular.

What is the equation of the director circle of a hyperbola?

The equation of the director circle of a hyperbola is x² + y² = a² - b².

Explain the condition for a line y = mx + c to be tangent to the hyperbola x²/a² - y²/b² = 1.

The condition for the line to be tangent is that the equation c² = a²m² - b² must hold, where c is the y-intercept.

Fill in the blank: The eccentricity e of a hyperbola is defined as ___

The eccentricity e of a hyperbola is defined as e = √(1 + b²/a²).

What is the equation of the tangent to the hyperbola at the point (x₁, y₁)?

The equation of the tangent at the point (x₁, y₁) on the hyperbola x²/a² - y²/b² = 1 is given by x x₁/a² - y y₁/b² = 1.

Describe the concept of the chord of contact for a hyperbola.

The chord of contact for a point (x₁, y₁) with respect to the hyperbola x²/a² - y²/b² = 1 is given by the equation T = 0, where T is the equation formed using the point and the hyperbola.

What is the significance of the lengths of the transverse and conjugate axes in defining the shape of a hyperbola?

The lengths of the transverse axis (2a) and the conjugate axis (2b) determine the spread of the hyperbola. The transverse axis defines the direction of the hyperbola's opening, while the conjugate axis defines how 'wide' it appears.