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Find the equation of the parabola with vertex (0,0) and focus (8,0).

a) y² = 8x
b) x² = 8y
c) y² = -8x
d) x² = -8y

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Final answer:

The equation of the parabola with the given vertex and focus is y² = 8x, which is option a).

Step-by-step explanation:

The equation of a parabola with vertex at the origin and focus on the x-axis can be derived using the standard form of a parabola that is symmetric about the x-axis, which is y² = 4px, where p is the distance from the vertex to the focus.

In this case, the focus is at (8,0), so p is 8. Therefore, by plugging in the value of p, we get the equation y² = 4*8*x, which simplifies to y² = 32x. Among the given options, the one that matches this form is a) y² = 8x.

A parabola is a U-shaped curve that is defined by a specific mathematical equation and has several distinctive properties. It is a type of conic section and is characterized by its symmetry, vertex, and axis.

Parabolas have many real-world applications and are commonly found in physics (such as projectile motion), engineering (designing reflectors or antennas), and mathematics, where they're fundamental in algebra, calculus, and geometry.

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