What are the key properties of a standard rightward opening parabola?

Focus at (a, 0), directrix x = -a, symmetric about x-axis.

How can a leftward opening parabola be described?

Flipped about y-axis, same properties as rightward parabola.

Focal chords passing through focus have reciprocals, product = 1.

(at^2, 2at), 4a is length of lattice rectum.

Point, parametric, or slope form, specific methods for each.

Understanding the properties of parabolas, including the focus at (a, 0) and directrix x = -a, is essential for mastering conic sections in GE exams, with parametric coordinates (at^2, 2at) providing additional insight into the structure of these curves.
Delving into hyperbolas, recognizing the various forms of tangent and normal equations, such as point, parametric, and slope forms, is crucial for solving problems related to these geometric figures, emphasizing the interconnected nature of conic sections and the importance of mastering their properties for academic success.

What are the key properties of a standard rightward opening parabola?
Focus at (a, 0), directrix x = -a, symmetric about x-axis.
How can a leftward opening parabola be described?
Flipped about y-axis, same properties as rightward parabola.
What is the relationship between focal chords and their parameters on a parabola?
Focal chords passing through focus have reciprocals, product = 1.
What are the parametric coordinates for a point on a parabola?
(at^2, 2at), 4a is length of lattice rectum.
How is the equation of a tangent to a parabola represented?
Point, parametric, or slope form, specific methods for each.

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Conic sections are crucial for GE exams, with students struggling to remember numerous formulas and equations.
Parabolas are a key focus, with standard rightward and leftward opening parabolas being commonly asked about.
The equation for a standard rightward opening parabola is y^2 = 4ax, with a representing the distance between the vertex and focus/directrix.
The focus coordinates are (a, 0) and the directrix equation is x = -a, with the parabola being symmetric about the x-axis.
Parametric coordinates for any point on the parabola are (at^2, 2at), with 4a representing the length of the lattice rectum.
Flipping the parabola about the y-axis results in a leftward opening parabola, with the same properties as the rightward one.
Focal chords passing through the focus have reciprocals of their parameters, with the product always equal to 1.
The length of a focal chord is a(t + 1/t)^2, with the lattice rectum being the smallest focal chord at 4a.
The equation of a tangent to a parabola can be in point, parametric, or slope form, with specific methods for each.
Moving on to ellipses, the standard horizontal ellipse's equation is x^2/a^2 + y^2/b^2 = 1, with key relationships between a, b, and eccentricity e.

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Conjugate hyperbola intersects the y-axis, represented by x^2/a^2 - y^2/b^2 = -1, indicating a vertical hyperbola with upward and downward branches.
Equation of tangent for a hyperbola has three forms: point, parametric, and slope, with the parametric coordinates for a point on a standard horizontal hyperbola being a sec Theta, beta and Theta.
The slope form for a tangent to a standard horizontal hyperbola is y = mx + c, where c = ±√(a^2m^2 - b^2), resulting in two parallel tangents with the same slope.
The equation of normal for a hyperbola also has three forms, derived from the equation of tangent, with the slope of the normal being the negative reciprocal of the tangent's slope at x1, y1.