Problem 73 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(P\) be a point on a ladder of length \(a+b, P\) being \(a\) units from the top end. As the ladder slides with its top end on the \(y\)-axis and its bottom end on the \(x\)-axis, \(P\) traces out a curve. Find the equation of this curve.

Step-by-Step Solution

Answer

The curve traced by point \(P\) is an ellipse: \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\).

Step 1: Understand the Problem

We have a ladder of length \(a+b\), with point \(P\) on it at a distance \(a\) from the top. As the ladder slides, \(P\) traces a curve, and we need to find the equation of this curve.

Step 2: Set Up the Triangle

Consider the ladder in a coordinate system where the top end is at \((0,y)\) and the bottom end is at \((x,0)\). The length of the ladder is the hypotenuse of a right triangle with legs on the \(x\)- and \(y\)-axes, so \(x^2 + y^2 = (a+b)^2\).

Step 3: Position of Point P

Let \(P\) have coordinates \((x_P, y_P)\). Since \(P\) is \(a\) units from the top end and \(b\) units from the bottom end, the positions \((x_P, y_P)\) can be defined using similar triangles: \[ x_P = \frac{b}{a+b}x \] and \[ y_P = \frac{a}{a+b}y. \]

Step 4: Eliminate Parameters

From Step 2, we have \(x^2 + y^2 = (a+b)^2\). Substituting \(x = \frac{a+b}{b}x_P\) and \(y = \frac{a+b}{a}y_P\) into this equation, we get: \[ \left( \frac{a+b}{b}x_P \right)^2 + \left( \frac{a+b}{a}y_P \right)^2 = (a+b)^2.\]

Step 5: Simplify the Equation

Divide both sides of the equation from Step 4 by \((a+b)^2\): \[ \frac{x_P^2}{b^2} + \frac{y_P^2}{a^2} = 1.\] This is the equation of an ellipse centered at the origin, with semiaxes of lengths \(b\) and \(a\) on the x- and y-axes respectively.

Key Concepts

Ladder Problem

The Ladder Problem is a classic exercise in calculus that enhances your understanding of motion and geometry using intuitive principles. Imagine a straight ladder positioned with one end on the y-axis and the other on the x-axis. The ladder slides uniformly, forming a right triangle with both coordinate axes. Point P, placed at a specific distance from the top of the ladder, traces a distinct path as it slides.

• The ladder's length is constant, forming the hypotenuse of the triangle.
• As the ladder slides, both x and y coordinates change.
• By understanding this motion, we visualize how point P moves.

Grasping this setup is key to visualizing the kind of curve point P traces on this constant motion of the ladder.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, merges algebra and geometry using a coordinate system. It allows us to represent geometric figures as algebraic equations. In the Ladder Problem, coordinate geometry is at play as the ladder slides. This involves placing the ladder within a coordinate system where:

• Its top and bottom ends are fixed on axes causing x and y coordinates to change with time.
• The relationship between these coordinates is essential to forming the right triangle.
• The length of the ladder remains constant, described by the formula:
  \[ x^2 + y^2 = (a+b)^2 \]

Coordinate geometry facilitates understanding how each part of the ladder contributes to the whole system and how the geometry leads us to a tangible equation of the trajectory point P follows.

Parametric Equations

Parametric Equations are crucial in modeling scenarios where two or more variables change with respect to another variable, often time. In the Ladder Problem, the movement of point P is described using parameters that change as the ladder slides.

• We express point P's position in terms of the ladder's movement on x and y axes.
• This involves assigning parameters to its motions:
  \[ x_P = \frac{b}{a+b}x \text{ and } y_P = \frac{a}{a+b}y \]
• Eliminating these parameters involves substituting to unveil the equation of the ellipse for P's path.

The use of parametric equations simplifies the complexity of motion, leading to an elegant representation where P draws an elliptical path on a coordinate plane. These equations are powerful tools in connecting physics and geometry, forming the basis of the mathematical expression of the curve made by point P.