Problem 3 - Algebra and Trigonometry

Question

Write each equation in terms of a rotated \(x^{\prime} y^{\prime}-\)system using \(\theta,\) the angle of rotation. Write the equation involving \(x^{\prime}\) and \(y^{\prime}\) in standard form. $$x^{2}-4 x y+y^{2}-3=0 ; \theta=45^{\circ}$$

Step-by-Step Solution

Answer

The equation of the given system when rotated by \(45^{\circ}\) and then written in standard form is \(x^{\prime 2} - y^{\prime 2} = 3\).

Step 1: Transform the equation using the rotation formulas

Use the rotation formulas on the given equation. So, you will replace \(x\) and \(y\) with \(x^{\prime} = x \cos \theta - y \sin \theta\) and \(y^{\prime} = x \sin \theta + y \cos \theta\) respectively. In this case, \(\theta=45^{\circ}\) which is equivalent to \(\pi/4\) in radians. Doing the substitution and simplifying gives \(x^{\prime 2} - y^{\prime 2} - 3 = 0\).

Step 2: Convert to Standard Form

The above equation needs to be in standard form, for which, the coefficient of \(x^{\prime 2}\) and \(y^{\prime 2}\) must be equal and the constant should be on the other side of the equation. Thus, the standard form of the equation will be \(x^{\prime 2} - y^{\prime 2} = 3\).

Key Concepts

Angle of Rotation

The angle of rotation is an important concept in coordinate geometry. It involves rotating the entire coordinate system around the origin by a specified angle, here denoted by \(\theta\). When we talk about \(\theta\), it is the angle between your original axis and the new, rotated axis. In this exercise, \(\theta = 45^{\circ}\), which means we're rotating the axes by 45 degrees.

This rotation transforms the coordinates \((x, y)\) into a new system \((x', y')\). Understanding this concept is crucial as it helps solve equations and analyze shapes from different perspectives. It's like looking at a problem from a new angle to gain new insights.

When rotating, the common choice is to rotate counterclockwise, but make sure to follow the specific instructions given. It affects how the rotation formulas are applied, which directly changes the coordinate expressions.

Standard Form Equation

After transforming an equation using rotation, it's essential to convert it into a standard form. The standard form of an equation typically involves simplifying it to a recognizable format, making equations easier to interpret and analyze.

In this exercise, our original equation becomes \(x'^{2} - y'^{2} - 3 = 0\) after applying the rotation formulas. To put it into standard form, you'll rearrange this to \(x'^{2} - y'^{2} = 3\). This way, both variables are on one side, and the constant is on the other.

Simplifying equations into standard form not only helps to recognize conic sections such as ellipses, parabolas, and hyperbolas more easily but also facilitates finding other important properties and elements faster.

Rotation Formulas

Rotation formulas are vital for changing coordinates from one set of axes to another. They are used when an equation or a shape is rotated about the origin. Two new variables \( x' \) and \( y' \) are introduced to represent the axes of the rotated system, and they are defined using the following formulas:

• \( x' = x \cos \theta - y \sin \theta \)
• \( y' = x \sin \theta + y \cos \theta \)

These formulas are derived from trigonometric principles of rotation. They tell you how much to "move" \(x\) and \(y\) in the new direction specified by \(\theta\). In our task, with an angle \(\theta = 45^{\circ}\), you would use these formulas to transition from \((x, y)\) to \((x', y')\).

Applying rotation formulas allows you to see the effects of rotation directly on the shape or figure represented by the initial equation. It's like rotating your view-point to understand an object or function differently.