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Magazine Chapter 10: Problem 164
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. $$y^{2}-24 x+4 y-68=0$$
Short Answer
Expert verified
The vertex is \((-3, -2)\), the focus is \((3, -2)\), and the directrix is \(x = -9\).
Step by step solution
01
Rearrange Terms
The given equation is \(y^2 - 24x + 4y - 68 = 0\). Start by rearranging the terms so all terms involving \(y\) are on one side and constants on the other. We get \(y^2 + 4y = 24x + 68\).
02
Complete the Square for y
To complete the square for \(y\), take the coefficient of \(y\), which is \(4\), divide it by 2 to get \(2\), and then square it to get \(4\). Add and subtract this square inside the equation: \(y^2 + 4y + 4 = 24x + 68 + 4\). This becomes \((y + 2)^2 = 24x + 72\).
03
Simplify Equation
Simplify the equation to \((y + 2)^2 = 24(x + 3)\). This is the equation of a parabola in standard form \((y - k)^2 = 4p(x - h)\).
04
Identify Vertex
From the standard form \((y - k)^2 = 4p(x - h)\), we identify the vertex \((V)\) as \((h, k)\). Thus, \(V = (-3, -2)\).
05
Determine the Focus
The value \(4p = 24\) implies \(p = 6\). Since \(p\) is positive and the equation is \(x - h\), this parabola opens to the right. The focus \((F)\) is \((h + p, k)\), so \(F = (3, -2)\).
06
Determine the Directrix
For a parabola opening to the right, the directrix is a vertical line \(x = h - p\). Therefore, the directrix \((d)\) is \(x = -9\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used to simplify quadratic expressions. It allows us to change the appearance of a quadratic by creating a perfect square trinomial. This is particularly helpful in rewriting parabolic equations into a more manageable form known as "standard form."
- First, identify the coefficient of the linear term. In the example provided, for the term involving \(y\), this coefficient is \(4\).
- Next, divide the coefficient by \(2\) and then square the result. Here, \(4\) divided by \(2\) equals \(2\), and \(2\) squared is \(4\).
- Add and subtract this square to the equation: originally \(y^2 + 4y\) becomes \(y^2 + 4y + 4 - 4\). This can be rewritten as \((y + 2)^2\).
Standard Form of a Parabola
The standard form of a parabola provides a straightforward way to analyze and graph the parabola. It takes the form \[(y - k)^2 = 4p(x - h)\]when the parabola opens horizontally as in our example.
- The terms \(h\) and \(k\) are the coordinates of the vertex of the parabola.
- The term \(4p\) relates to the focal width of the parabola, determining how "wide" it opens.
- The direction in which the parabola opens (left, right, up, down) can be determined by the position of \(x\) and \(y\) in the equation.
Vertex of a Parabola
The vertex of a parabola is a crucial point and can be thought of as the point where the parabola changes direction. For the standard form equation \[(y - k)^2 = 4p(x - h),\]the vertex is given by the point \((h, k)\).
- In our exercise, after rearranging and completing the square, we have \((y + 2)^2 = 24(x + 3)\).
- Here, the vertex \((h, k)\) can be identified as \((-3, -2)\).
Focus of a Parabola
The focus of a parabola is a fixed point that helps determine the shape and direction of the parabola. It can be directly extracted once the parabola is in standard form.
- The distance of the focus from the vertex is given by the value of \(p\). From \(4p = 24,\) we find \(p = 6\).
- Since the standard form is \((y - k)^2 = 4p(x - h)\), the parabola opens to the right (positive \(p\)).
- The coordinates of the focus are calculated by adding \(p\) to the \(h\)-coordinate of the vertex, so \((h + p, k) = (3, -2)\).
Directrix of a Parabola
The directrix of a parabola is a line perpendicular to the axis of symmetry and serves as a reference from which the parabola reflects. It plays a role opposite to the focus.
- For a parabola opening horizontally, like \((y - k)^2 = 4p(x - h)\), the directrix is a vertical line.
- The equation of the directrix can be derived by subtracting \(p\) from the \(h\)-coordinate: \(x = h - p\).
- In our example, with \(h = -3\) and \(p = 6\), the directrix is \(x = -9\).
