Problem 22 - Analytical Geometry: 2D and 3D

Question

Let a line \(L\) has intercepts \(a\) and \(b\) on the coordinate axes. When the axes are rotated through an angle, keeping the origin fixed, the same line \(L\) has intercepts \(p\) and \(q\). Obtain the relation between \(a, b, p\), and \(q\).

Step-by-Step Solution

Answer

Short Answer: The relation between the intercepts a, b, p, and q after rotating the coordinate system through an angle θ is ab = pq(cos²(θ) + sin²(θ)).

Step 1: Find the slope and equation of the line in the initial coordinate system

Given that the line L has intercepts a and b on the coordinate axes, we can find the slope of the line by using the formula: Slope (m) = -b/a Now we can write the equation of the line L using the slope-intercept form: y = mx + b

Step 2: Find the angle of rotation

Let the angle of rotation be denoted by \(\theta\). We will use this angle in later steps to find the new intercepts p and q.

Step 3: Rotate the coordinate system and find the new slope and equation of the line in the rotated coordinate system

After rotating the coordinate axes through an angle \(\theta\), the new coordinates of the points (a,0) and (0,b) are transformed to (a*cos(\theta), a*sin(\theta)) and (-b*sin(\theta), b*cos(\theta)), respectively. Now, we can find the new slope of the line in the rotated coordinate system: New slope (m') = (b*cos(\theta) - a*sin(\theta))/(a*cos(\theta) + b*sin(\theta)) With the new slope, we can write the equation of the line L in the rotated coordinate system: y' = m'x' + q

Step 4: Find the relation between a, b, p, and q

We know that the intercepts of the line L in the rotated coordinate system are p and q. By substituting x'=0 and y'=0 in the equation of the line L in the rotated coordinate system, we can find the relations between a, b, p, and q. When x'=0: y' = m'x' + q = q 0 = -p*sin(\theta) + p*cos(\theta)*m' When y'=0: x' = p 0 = a*cos(\theta) + a*sin(\theta)*m' Now, by solving this system of equations simultaneously, we can obtain the relation between a, b, p, and q. The final relation is: ab = pq(cos^2(\theta) + sin^2(\theta))

Key Concepts

Coordinate Rotation

Coordinate rotation is a vital concept in analytical geometry, especially when transforming a geometric figure in the coordinate plane. When we rotate the axes around the origin by a certain angle, the coordinates of any point change based on the angle of rotation. This transformation is mathematically described using trigonometric functions, sine, and cosine. The basic idea is to see how points initially described in a standard coordinate system transform to a new position with a different orientation.
During a rotation by an angle \( \theta \), a point initially at \( (x, y) \) will move to a new position determined by the following transformations:

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

This transition helps in recalculating spatial attributes, like line intercepts, when the coordinate system is rotated.

Line Intercepts

In the context of a coordinate geometry, line intercepts refer to the points where a line intersects the coordinate axes. The x-intercept is the point where the line crosses the x-axis, while the y-intercept is where it crosses the y-axis. For a given line with a slope-intercept form equation of \( y = mx + c \), the intercepts are:

• x-intercept: \( -c / m \)
• y-intercept: \( c \)

Understanding intercepts is important as they provide a straightforward way to graphically represent a line and solve intersections with the axes. In the problem context, the line initially has intercepts \( a \) and \( b \), which means the line crosses the x-axis at \( (a, 0) \) and the y-axis at \( (0, b) \). These values change when the coordinate system is rotated, giving new intercepts \( p \) and \( q \) in the rotated system.

Equation of a Line

The equation of a line is a fundamental concept in coordinate geometry. It provides a mathematical description of a straight line on the x-y plane. The most common form is the slope-intercept form, \( y = mx + c \), where:

• \( m \) is the slope of the line
• \( c \) is the y-intercept

To find the equation of a line from given intercepts \(a\) and \(b\), we use the equation \( y = mx + b \) adjusting for initial intercepts. After rotation, thanks to coordinate rotation formulas, we reconceive the equation based on the new orientation of the axis. This involves transforming our initial description using trigonometric identities until a clear new equation emerges.

Angle of Rotation

The angle of rotation is an essential element in understanding how geometric shapes and lines transform in the coordinate plane. Denoted by \( \theta \), it represents the angle by which the entire coordinate system is rotated around a specific point, usually the origin. The rotation angle affects the trigonometric identities used to determine new coordinates and intercepts:

• The cosine component \( \cos(\theta) \) affects horizontal changes.
• The sine component \( \sin(\theta) \) affects vertical changes.

In analytical geometry, rotating the axes retains the original properties within the constraints of trigonometric transformations. Such an angle is pivotal in correlating original and transformed intercepts on the coordinate axes.