Problem 35 - Thomas Calculus

Question

What effect does a \(90^{\circ}\) rotation about the origin have on the equations of the following conic sections? Give the new equation in each case. a. The ellipse \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1 \quad(a>b)\) b. The hyperbola \(\left(x^{2} / a^{2}\right)=\left(y^{2} / b^{2}\right)=1\) c. The circle \(x^{2}+y^{2}=a^{2}\) d. The line \(y=m x \) e. The line \(y=m x+b\)

Step-by-Step Solution

Answer

The equations after rotation are: Ellipse: \\( \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \\); Hyperbola: \\( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \\); Circle remains \\( x^2 + y^2 = a^2 \\); Line y=mx becomes \\( y = -\frac{1}{m}x \\); Line y=mx+b becomes \\( y = -\frac{1}{m}x + \frac{b}{m} \\).

Step 1: Understanding 90-degree Rotation

A 90-degree rotation about the origin in a Cartesian plane involves swapping the coordinates and changing the sign of the new x-coordinate. If a point is originally \(x, y\), after a 90-degree rotation it becomes \(-y, x\). This concept will be applied to all the given equations.

Step 2: Ellipse Rotation

The equation of the ellipse is \left(\frac{x^{2}}{a^{2}}\right)+\left(\frac{y^{2}}{b^{2}}\right)=1\. By substituting each \(x, y\) with \(-y, x\), the new equation becomes \left(\frac{(-y)^2}{a^{2}}\right)+\left(\frac{x^{2}}{b^{2}}\right)=1\ or simplifying to \left(\frac{y^{2}}{a^{2}}\right)+\left(\frac{x^{2}}{b^{2}}\right)=1\.

Step 3: Hyperbola Rotation

The equation of the hyperbola is \left(\frac{x^{2}}{a^{2}}\right)=\left(\frac{y^{2}}{b^{2}}\right)=1\. Applying the rotation \(x, y \to -y, x\), the equation becomes \left(\frac{(-y)^{2}}{a^{2}}\right)-\left(\frac{x^{2}}{b^{2}}\right)=1\, which simplifies to \left(\frac{y^{2}}{a^{2}}\right)-\left(\frac{x^{2}}{b^{2}}\right)=1\.

Step 4: Circle Rotation

For the circle represented by \(x^{2}+y^{2}=a^{2}\), rotation does not change its equation. Replacing \(x, y \to -y, x\) results in \((-y)^{2}+x^{2}=a^{2}\), which simplifies to \(y^{2}+x^{2}=a^{2}\). Thus, the equation remains the same.

Step 5: Line y = mx Rotation

For the line \(y=mx\), substitute \(x, y \to -y, x\) to get \(x = m(-y)\) or \(x = -my\). The equation can be rewritten as \y = -\frac{1}{m}x\.

Step 6: Line y = mx + b Rotation

Substitute \(x, y \to -y, x\) into \(y = mx + b\) to get \(x = m(-y) + b\) which simplifies to \(x = -my + b\). Rewriting, we have \y = -\frac{1}{m}x + \frac{b}{m}\.

Key Concepts

Coordinate Rotation

When we talk about coordinate rotation, we mean turning the entire coordinate system around a fixed point, typically the origin. In a 90-degree rotation around the origin in a Cartesian plane, each point \(x, y\) transforms to \(-y, x\). This has interesting effects on the equations of conic sections.:

• Inverts and repositions shapes on the graph.
• Helps analyze symmetrical properties.
• Keeps distances from the origin the same, conserving their geometric properties.

The rule of swapping and reversing coordinates is essential for correctly changing the equations of shapes like ellipses, hyperbolas, and lines. This transformation maintains their overall structure but alters their orientation on the plane.

Ellipse Transformation

Ellipses are a type of conic section, characterized by the equation \(\left( \frac{x^{2}}{a^{2}} \right) + \left( \frac{y^{2}}{b^{2}} \right) = 1\). Rotating an ellipse by 90 degrees changes its principal axes, effectively swapping its horizontal and vertical radii. Here's how:

• Original equation: \(\left( \frac{x^{2}}{a^{2}} \right) + \left( \frac{y^{2}}{b^{2}} \right) = 1\).
• After substituting \(x, y o -y, x\), the equation becomes \(\left( \frac{y^{2}}{a^{2}} \right) + \left( \frac{x^{2}}{b^{2}} \right) = 1\).

This rotation changes the focus of the ellipse while maintaining the equation's structure. It highlights the inherent symmetry and can help visualize the ellipse's orientation in space, which is useful for practical applications and theoretical analysis.

Hyperbola Transformation

Hyperbolas, another kind of conic section, have an equation like \(\left( \frac{x^{2}}{a^{2}} \right) - \left( \frac{y^{2}}{b^{2}} \right) = 1\). Rotating a hyperbola affects its asymptotes and the orientation of its branches. Here's the process:

• Original hyperbola: \(\left( \frac{x^{2}}{a^{2}} \right) - \left( \frac{y^{2}}{b^{2}} \right) = 1\).
• After rotation to \(-y, x\), it becomes \(\left( \frac{y^{2}}{a^{2}} \right) - \left( \frac{x^{2}}{b^{2}} \right) = 1\).

The rotation flips the direction of the hyperbola's opening and modifies its asymptotic lines. This knowledge is crucial for graphing and understanding hyperbolas' behavior when subjected to linear transformations.

Circle Properties

A circle's equation is simply \(x^{2} + y^{2} = a^{2}\), where \(a\) is the radius. Interestingly, rotating a circle by any angle, including 90 degrees, leaves its equation unchanged. This occurs because:

• A circle is perfectly symmetrical around the origin.
• All points remain equidistant from the center, ensuring consistent properties regardless of orientation.

The circle's uniformity makes it resilient to transformations that would typically alter other shapes. Understanding this helps appreciate the circle’s symmetry and why it appears identical from every angle.

Linear Transformation

Linear transformations, such as those involving lines, can alter their slope and y-intercept. Consider the line \(y = mx + b\). A 90-degree rotation changes its properties significantly:

• With \(y = mx\), rotating yields \(y = -\frac{1}{m}x\).
• For \(y = mx + b\), the rotated form is \(y = -\frac{1}{m}x + \frac{b}{m}\).

These transformations demonstrate how lines can be re-oriented within the plane. The flipped slope indicates a change in direction, while the adjusted intercept reflects the line’s new position relative to the axes. Recognizing these shifts is fundamental in linear algebra and geometry when translating between various coordinate systems.