Question

when graphing a parabola, the focus is defined as a . located the parabola, and the directrix is defined as a located the parabola. they are both units away from the vertex. vector line plane point
using cramer’s rule to solve the following system of . which would you use for the denominator of each variable?

Explanation:

To solve the problem about the focus and directrix of a parabola:

For the focus:

• A parabola's focus is a point (from the options: vector, line, plane, point) located inside the parabola, at a distance of \( \frac{1}{4|a|} \) units from the vertex (where \( a \) is the coefficient of \( x^2 \) or \( y^2 \) in the parabola’s equation).

For the directrix:

• The directrix is a line (from the same options) located outside the parabola, also at a distance of \( \frac{1}{4|a|} \) units from the vertex, and parallel to the axis of symmetry (perpendicular to the parabola’s opening direction).

Final Answer (filling the blanks):

• Focus: defined as a \(\boldsymbol{\text{point}}\), located \(\boldsymbol{\text{inside}}\) the parabola.
• Directrix: defined as a \(\boldsymbol{\text{line}}\), located \(\boldsymbol{\text{outside}}\) the parabola.
• They are both \(\boldsymbol{\frac{1}{4|a|}}\) units away from the vertex (where \( a \) is the coefficient of \( x^2 \) or \( y^2 \) in the parabola’s standard form).

• Focus: By definition, the focus of a parabola is a fixed point inside the curve, equidistant from the vertex and the directrix.
• Directrix: The directrix is a fixed line outside the parabola, perpendicular to the axis of symmetry, and equidistant from the vertex as the focus.
• Distance: For a parabola in the form \( y = ax^2 + bx + c \) (or \( x = ay^2 + by + c \)), the distance from the vertex to the focus (and to the directrix) is \( \frac{1}{4|a|} \) units.

Answer:

• Focus: By definition, the focus of a parabola is a fixed point inside the curve, equidistant from the vertex and the directrix.
• Directrix: The directrix is a fixed line outside the parabola, perpendicular to the axis of symmetry, and equidistant from the vertex as the focus.
• Distance: For a parabola in the form \( y = ax^2 + bx + c \) (or \( x = ay^2 + by + c \)), the distance from the vertex to the focus (and to the directrix) is \( \frac{1}{4|a|} \) units.