Problem 162 For the following exercises, rew... [FREE SOLUTION]

Chapter 10: Problem 162

For the following exercises, rewrite the given equation in standard form, and then determine the vertex $(V),$ focus $(F)$, and directrix $(d)$ of the parabola. $$x^{2}-4 x-24 y+28=0$$

Short Answer

Expert verified

Vertex: (2,1), Focus: (2,7), Directrix: y=-5.

Step by step solution

Rearrange the Equation

Start by rearranging the equation so that the terms involving $x$ are isolated on one side. Given equation is $x^2 - 4x - 24y + 28 = 0$. Move the terms with $y$ to the right side: $x^2 - 4x = 24y - 28$.

Complete the Square

To rewrite the quadratic expression $x^2 - 4x$ in a form that makes completing the square possible, we add and subtract the square of half the coefficient of $x$.The coefficient of $x$ is $-4$, so half of this is $-2$. Square this to get $((-2)^2) = 4$.Add and subtract 4 inside the equation:$x^2 - 4x + 4 - 4 = 24y - 28$, which simplifies to $(x - 2)^2 - 4 = 24y - 28$.

Simplify and Solve for y

Add 4 to both sides to isolate the complete square term:$(x - 2)^2 = 24y - 24$.Now, simplify the right side:$(x - 2)^2 = 24(y - 1)$.

Identify the Standard Form

The equation $(x - 2)^2 = 24(y - 1)$ is now in the standard form of a vertical parabola, which is $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex, and $p$ is the distance from the vertex to the focus or directrix.

Determine the Vertex (V)

From the equation $(x - 2)^2 = 24(y - 1)$, we see that the vertex form gives us $h = 2$ and $k = 1$. Therefore, the vertex $V$ is at $(2, 1)$.

Calculate the Focus (F)

In the equation $(x - 2)^2 = 24(y - 1)$, $4p = 24$, so $p = \frac{24}{4} = 6$. The parabola opens upwards, so the focus is at $(h, k + p)$, which is $(2, 1 + 6) = (2, 7)$.

Find the Directrix (d)

The directrix is a horizontal line $p$ units below the vertex because the parabola opens upwards. Therefore, the equation of the directrix is $y = k - p = 1 - 6 = -5$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form

The standard form of a parabola is a way to express the equation for easy identification of its properties, such as the vertex, focus, and directrix. For a parabola that opens vertically, the standard form is

$(x - h)^2 = 4p(y - k)$.

Here:

$(h, k)$ is the vertex of the parabola.
$p$ is the distance from the vertex to the focus or directrix.

This form allows us to quickly identify the key features of the parabola by examining the coefficients. In the given exercise, we transformed the equation

$x^2 - 4x - 24y + 28 = 0$

into the standard form

$(x - 2)^2 = 24(y - 1)$.

Notice that completing the square is an important technique to achieve this form, as it helps in isolating the squared term and arranging the remaining parts into the desirable structure. Understanding standard form opens up all the other components of the parabola for exploration.

Vertex

The vertex of a parabola is a pivotal point and can give us a lot of information about the graph.

It represents the minimum or maximum point of a parabola, depending on its orientation.
For a quadratic function in standard form $(x - h)^2 = 4p(y - k)$,

the vertex is clearly identified as

$(h, k)$.

In our problem, the vertex was determined to be at

$(2, 1)$

by examining the equation after rewriting it in standard form.This point is where the parabola changes direction.
Understanding the vertex's location helps in sketching the basic shape of the parabola and gives insights into other features like symmetry and orientation.

Focus

The focus of a parabola is an essential concept in understanding its reflective properties. The focus sits inside the parabola and can be found at a specific distance from the vertex.

In the standard form, $(x - h)^2 = 4p(y - k)$,
we use the value of $p$ to find the focus.

Here, the distance $p$ is calculated to be $6$, so the focus is

$(h, k + p) = (2, 7)$.

This position allows a parabola to have a property where any ray parallel to the axis of symmetry bounces off the parabola and passes through the focus. This concept is used in various applications like satellite dishes and flashlight reflectors where focus and reflectivity are vital.

Directrix

The directrix of a parabola is a line that, along with the focus, helps define the curve geometrically. It is always perpendicular to the axis of symmetry of the parabola.

For a parabola with an equation in standard form, $(x - h)^2 = 4p(y - k)$,
the directrix is $p$ units "opposite" to the focus.

In our case, the directrix is found at the equation

$y = k - p = -5$.

This means the directrix provides a reference line where the distance from any point on the parabola to the focus equals the distance to this line.Understanding the concept of a directrix is crucial in mastering the geometric definition of a parabola, completing the set of elements that describe the curve's shape in a plane.

91影视

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F)\), and directrix \((d)\) of the parabola. $$x^{2}-4 x-24 y+28=0$$

Short Answer

Step by step solution

Rearrange the Equation

Complete the Square

Simplify and Solve for y

Identify the Standard Form

Determine the Vertex (V)

Calculate the Focus (F)

Find the Directrix (d)

Key Concepts

Standard Form

Vertex

Focus

Directrix

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