A parabola in the standard (x, y) coordinate plane is defined by the equation y^2 = 12x. W… — ACT

A parabola in the standard $(x, y)$ coordinate plane is defined by the equation $y^2 = 12x$ . What is the $x$ -coordinate of the focus of this parabola?

Answer: 3

Answer

The

x

-coordinate of the focus is 3.

The equation

y^2 = 12x

represents a parabola with vertex at the origin opening to the right. The standard form for such a parabola is

y^2 = 4px

, where the focus is located at

(p, 0)

. By setting

4p = 12

, we find

p = 3

. Thus, the focus is

(3, 0)

, and its

x

-coordinate is 3.

Step-by-Step Solution

Identify the standard form of the parabola's equation.

The equation

y^2 = 12x

fits the standard form of a horizontal parabola with its vertex at the origin,

y^2 = 4px

This allows us to relate the given equation to the coordinate of the focus, which is located at

(p, 0)

Solve for the parameter

p

4p = 12

, which gives

p = 3

By equating the coefficients of

x

from the given equation and the standard form, we can find the value of

p

Determine the focus coordinate.

The focus is at

(3, 0)

, so the

x

-coordinate is 3.

The focus of a parabola of the form

y^2 = 4px

has coordinates

(p, 0)

Key Concept

Focus of a Parabola

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