Question

Find the focus and directrix of the parabola $$ 2 y^{2}-4 y-10 x=0 $$

Short answer

Focus: \(\left(\frac{19}{20}, 1\right)\); Directrix: \(x = -\frac{29}{20}\).

Step-by-step solution

Rewrite the Equation in Standard Form

First, isolate the terms involving "y" on one side and those involving "x" on the other. We start with:\[ 2y^2 - 4y - 10x = 0 \].Add \(10x\) to both sides:\[ 2y^2 - 4y = 10x \].Now, divide each term by 2 to simplify:\[ y^2 - 2y = 5x \].This is the equation after separation into canonical form for a parabola.

Complete the Square

We need to complete the square for the \(y\) terms. Start with:\[ y^2 - 2y \].Take half of the coefficient of \(y\) (which is \(-2\)), square it and add & subtract it inside the equation:\[ y^2 - 2y + 1 - 1 \].This simplifies to:\[ (y - 1)^2 - 1 \].Thus, the equation becomes:\[ (y - 1)^2 = 5x + 1 \].

Rewrite Equation for Vertex Form

Rewrite the equation:\[ (y - 1)^2 = 5(x + 1/5) \].Thus, we wrote the equation in the form \((y - k)^2 = 4p(x - h)\) where \(k = 1\), \(h = -1/5\), and \(4p = 5\).

Identify the Values of h, k, and p

From the equation \((y - 1)^2 = 5(x + 1/5)\), let's identify:\[ k = 1, \, h = -1/5, \, 4p = 5 \].So, \( p = \frac{5}{4} \).

Determine the Focus of the Parabola

The focus of a parabola of the form \((y - k)^2 = 4p(x - h)\) is given by \((h + p, k)\).Substitute the values:\[ h = -1/5, \, k = 1, \, p = 5/4 \].Thus, the focus is:\[ \left(- \frac{1}{5} + \frac{5}{4}, 1\right) = \left(\frac{19}{20}, 1\right) \].

Determine the Directrix of the Parabola

The equation for directrix is \( x = h - p \).Using \( h = -1/5 \) and \( p = 5/4 \):\[ x = - \frac{1}{5} - \frac{5}{4} = - \frac{1}{5} - \frac{25}{20} \].Combine the fractions:\[ x = - \frac{4}{20} - \frac{25}{20} = - \frac{29}{20} \].The equation of the directrix is \(x = -\frac{29}{20}\).

Key concepts

Parabola

A parabola is a special curve you might have seen in math class. It's the graph of a quadratic function and is shaped like a bowl or a U. Every parabola has a symmetry axis, which means it looks the same on both sides of this line. Parabolas come from slicing through a cone at an angle parallel to one side of the cone. This is why we call them conic sections. In mathematic terms, parabolas are defined using quadratic equations, commonly expressed as either \( y = ax^2 + bx + c \) or \( x = ay^2 + by + c \).What makes a parabola different from other conic sections is how it has just one direction – it keeps opening wider as you move further along its length. Understanding how parabolas work, including their parts like focus and directrix, helps us solve many real-world problems, from satellite dishes focusing signals to the architecture of bridges.

Focus and Directrix

The concepts of focus and directrix are central to understanding a parabola. These are geometrical properties that help describe the shape and position of the parabola.

• Focus: The focus is a fixed point inside the parabola. All points on the parabola are equidistant from the focus and the directrix. This makes the focus a "reflection" point that influences the parabola’s shape and opens it.
• Directrix: The directrix is a line that is not part of the parabola but works together with the focus to guide its shape. Each point on the parabola is the same distance from the directrix as it is from the focus.

This means if you were to draw a line from any point on the parabola to the focus, it would be of equal length to a line drawn perpendicularly to the directrix. Understanding how these work helps in graphing parabolas and finding points that belong to them.

Completing the Square

Completing the square is a technique used in algebra to transform a quadratic equation into a form that makes it easy to graph or solve. This method is useful when you want to find the vertex of a parabola, which is a key point of the curve.In the context of our problem, starting with \( y^2 - 2y = 5x \):- We took half of the coefficient of \(y\), which was \(-2\), divided it to get \(-1\).- Squared \(-1\) to get \(1\) and added/subtracted it in the equation: \( y^2 - 2y + 1 - 1 \).This allowed us to group the terms into \((y - 1)^2\), simplifying the equation to more easily identify the vertex form which links directly to determining the parabola's focus and directrix.

Vertex Form

The vertex form of a parabola equation is particularly useful because it makes identifying the key parts of the parabola straightforward. The vertex form is given by either \((y - k)^2 = 4p(x - h)\) or \((x - h)^2 = 4p(y - k)\).In this specific setup:

• \((h, k)\) become the coordinates of the vertex – the point where the parabola changes direction, a tip or the "highest" or "lowest" point, depending on its orientation.
• The parameter \(4p\) determines the "width" or directness of the parabola's opening. A larger \(|4p|\) stretches the parabola wider.

Knowing how to rewrite a quadratic equation into the vertex form, like what was done in the given exercise, is crucial for functions, engineering, physics, and other applications requiring a precise understanding of parabolic paths and structures.