How to Find Rotation of Axes and General Form of Conic Sections

Step-by-step Guide to Find Rotation of Axes and General Form of Conic Sections

Here is a step-by-step guide to finding the rotation of axes and the general form of conic sections:

1. Rotation of Axes

Before diving into the conic sections, let’s understand what rotation of axes means. When you rotate the axes, you’re essentially looking at the same graph but from a different perspective, like turning your head to see something from a different angle.

Steps for Rotation of Axes:

1.1. Understand the Basics:

• Original Axes: \(x\) and \(y\)
• New Axes (rotated by an angle \(\theta\)): \(x′\) and \(y′\)

1.2. Equations Relating Old and New Axes: Given a point \(P \ (x,y)\) on the original axes, its coordinates on the new axes are \(P \ (x′,y′)\). The relationships are:

\(x=x′cos \ \theta−y′sin\ \theta\)

\(y=x′sin \ \theta+y′cos \ \theta\)

1.3. Using the Above Equations: To convert the equation of a curve from the old axes to the new axes, substitute the expressions for \(x\) and \(y\) from the relationships above.

2. General Form of Conic Sections

A conic section can be represented as:

\(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\)

Where, \(A, B, C, D, E\), and \(F\) are constants.

2.1. Identify the Conic Based on the General Form:

• Circle: \(B=0, A=C\)
• Ellipse: \(B=0, A≠C\)
• Parabola: \(B^2−4AC=0\)
• Hyperbola: \(B^2−4AC>0\)

2.2. Converting to Standard Form: If the equation contains the term \(xy\) (i.e., \(B≠0\)), it indicates that the conic is not aligned with the coordinate axes and needs to be rotated.

Steps to Convert to Standard Form:

Step 1: Divide the entire equation by the term that doesn’t contain \(x\) or \(y\) to make it equal to \(1\).

Step 2: For equations with \(xy\) terms, the angle of rotation \(\theta\) can be found using:

\(tan(2\theta)=\frac{B}{A−C}\)​ Calculate \(\theta\).

Step 3: Use the rotation of axes formulas mentioned earlier to transform the equation.

Step 4: The transformed equation will now be free of the \(xy\) term, placing it in one of the standard forms of conic sections (circle, ellipse, parabola, or hyperbola).

Final Word

the study of the rotation of axes helps in simplifying the representation of conic sections. By rotating the axes to eliminate the \(xy\) term, one can represent the conic section in its standard form, making it easier to identify and analyze.