Problem 34 - Algebra and Trigonometry

Question

(a) Use rotation of axes to show that the following equation represents a parabola. $$ 2 \sqrt{2}(x+y)^{2}=7 x+9 y $$ (b) Find the \(X Y\) - and \(x y-\) coordinates of the vertex and focus. (c) Find the equation of the directrix in \(X Y\) - and \(x y\) coordinates.

Step-by-Step Solution

Answer

The given equation represents a parabola. Vertex: \( (\frac{16}{21}, \frac{8}{21}) \) in xy-coordinates; Focus: \( (\frac{11}{21}, \frac{1}{21}) \). Directrix: \(X = \frac{41}{42}\).

Step 1: Identify Equation Components

The given equation is \(2\sqrt{2}(x+y)^2 = 7x + 9y\). We need to identify how this can be rewritten or transformed to recognize it as a parabola. The equation currently mixes \(x\) and \(y\).

Step 2: Rotate Axes

To remove mixed \(xy\) terms, we perform a rotation of axes. Given that the mixed term suggests a 45-degree rotation (because of \(x+y\)), we set \(X = \frac{x+y}{\sqrt{2}}\) and \(Y = \frac{x-y}{\sqrt{2}}\).

Step 3: Substitute for Rotation

Substitute \(x = \frac{X+Y}{\sqrt{2}}\) and \(y = \frac{X-Y}{\sqrt{2}}\) into the original equation. Simplifying both sides will confirm we've transformed to new axes properly.

Step 4: Simplify and Identify as Parabola

After substituting, simplify the equation to isolate terms and recognize it follows the standard form \((X - h)^2 = 4p(Y - k)\), which is the form of a parabola. Ensure no mixed terms remain.

Step 5: Find Vertex and Focus in New Axes

With the equation simplified, identify the vertex \((h, k)\) directly from the standard form. The focus will be \( \left(h, k + p \right) \). Translate back to original coordinates \((x, y)\).

Step 6: Find Directrix Equation

The directrix of the parabola is given by \(Y = k - p\). Substitute back to \(x,y\) coordinates, again using inverse rotation transformations to provide the directrix in original coordinates.

Key Concepts

Rotation of Axes

Sometimes, an equation isn't in the exact form we need for certain analyses, like identifying a conic section such as a parabola. In such cases, rotating the axes can be very handy. Rotation of axes is a mathematical trick to simplify an equation by getting rid of mixed terms, like the product of two different variables. This adjustment can reveal the underlying geometric shape more clearly.
To perform a rotation for an axis, we use transformation equations, usually in the form of substitutions. For example, if we suspect a 45-degree rotation will help, we might set:

• \(X = \frac{x+y}{\sqrt{2}}\)
• \(Y = \frac{x-y}{\sqrt{2}}\)

This change swaps our coordinate system around to eliminate mixed terms like \(xy\). By substituting these into the original equation, we simplify it into a form that's easier to analyze and identify, in this case, as a parabola.

Vertex of a Parabola

The vertex of a parabola is a crucial point. It's the "turning point" where the curve changes direction. In the standard parabola form \((X - h)^2 = 4p(Y - k)\), the vertex is straightforward to find. It's at the point \((h, k)\), which results directly from our simplified form.
In our problem, after the rotation and simplification, finding \((h, k)\) in the transformed coordinates gives us the vertex in the new \((X, Y)\) system. Transforming it back involves the inverse of the initial rotation. If you started with \(X = \frac{x+y}{\sqrt{2}}\)in your transformation, you'll substitute back with \(x = \frac{X + Y}{\sqrt{2}}\) and \(y = \frac{X - Y}{\sqrt{2}}\) to find the vertex in the original \((x, y)\) coordinates.

Focus of a Parabola

The focus of a parabola is another vital component of its geometric identity. It's a point located inside the curve and has special reflective properties. When a parabola is in the form \((X - h)^2 = 4p(Y - k)\), the focus is found at \((h, k + p)\), where \(p\) indicates how "open" or "wide" the parabola is.
Once we've identified \((h, k)\) and \(p\) from the standard equation, locating the focus becomes straightforward. This new point in \((X, Y)\) space must again be translated back to the original \((x, y)\) coordinates. Using the rotation transformation in reverse can help us pinpoint where the focus would appear in the original system.

Equation of the Directrix

Every parabola is complemented by a directrix, a line that exists outside the curve serving as a sort of boundary in tandem with the focus. The equation for the directrix when dealing with a transformed parabola is \(Y = k - p\). This equation represents a line parallel to the axis of symmetry of the parabola.
The directrix is as fundamental as the focus, offering another means to understand the parabola's orientation and geometry. Once obtained in the rotated \((X, Y)\) system, the directrix must be reverted into its counterpart in \((x, y)\) coordinates using the inverse transformations. This allows you to visualize the directrix in the context of the original coordinate plane, giving full understanding of both the parabola’s orientation and position.