What is the standard equation of a parabola with its vertex at the origin and focus at (a, 0)?

The standard equation of the parabola is y² = 4ax.

For the parabola defined by the equation y² = 4ax, what are the coordinates of the ends of the latus rectum?

The coordinates of the ends of the latus rectum are (a, 2a) and (a, -2a).

Fill in the blank: The distance of a point on the parabola from the focus is called the __________.

Focal distance.

True/False: The semi-latus rectum is the harmonic mean of the segments of a focal chord.

True. The semi-latus rectum is indeed the harmonic mean of the segments of a focal chord.

What is the relationship between the parameters t₁ and t₂ for the ends of a focal chord in a parabola?

For the ends of a focal chord (at, 2at₁) and (at², 2at₂), the relationship is t₁t₂ = -1.

Write the equation of the directrix for the parabola with focus at (a, 0).

The equation of the directrix is x = -a.

What is the length of the latus rectum for a parabola given by the equation y² = 4ax?

The length of the latus rectum is 4a units.

Fill in the blank: The parametric equations for the parabola y² = 4ax are __________.

X = at², y = 2at.

What is the equation of the pair of tangents from an external point P(x₁, y₁) to the parabola y² = 4ax?

The equation is SS₁ = T², where S: y² - 4ax, S₁: y² - 4ax₁, and T: yy₁ - 2a(x + x₁).

True/False: The parabola defined by the equation x² = 4ay opens upwards.

True. The parabola defined by the equation x² = 4ay opens upwards.

What is the parametric representation of the parabola defined by the equation x² = -4ay?

The parametric equations are x = -2at, y = -at².

Fill in the blank: The axis of the parabola y² = 4ax is the line __________.

Y = 0.

What is the standard equation of a parabola with its vertex at the origin and focus at (a, 0)?

The standard equation of the parabola is y² = 4ax.

If a parabola has an equation x² = 8y, what is the length of its latus rectum?

The length of the latus rectum is 4a = 8, so it is 8 units.

True/False: The focal distance of any point on a parabola is equal to its perpendicular distance from the directrix.

True. This follows from the definition of a parabola, which states that every point on a parabola is equidistant from the focus and the directrix.