Problem 78 - Calculus: Graphical, Numerical, Algebraic

Question

Limits and Geometry Let \(P \left( a , a ^ { 2 } \right)\) be a point on the parabola \(y = x ^ { 2 } , a > 0 .\) Let \(O\) be the origin and \(( 0 , b )\) the \(y\) -intercept of the perpendicular bisector of line segment \(O P .\) Find \(\lim _ { P \rightarrow O } b\)

Step-by-Step Solution

Answer

The limit of b as P approaches O is -1/2.

Step 1: Find the Slope of Line OP

The slope of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = (y_2 - y_1) / (x_2 - x_1)\). Here, \((x_1, y_1)=(a, a^2)\) and \((x_2, y_2)=(0, 0)\), so the slope of line OP is \(m_{OP} = (0 - a^2) / (0 - a) = -a\)

Step 2: Calculate the Slope of the Perpendicular Bisector

The slope of a line perpendicular to another line with slope \(m\) is \(-1/m\). Therefore, the slope of the perpendicular bisector is \(m_{bisector} = -1 / m_{OP} = 1/a\)

Step 3: Construct Equation of the Perpendicular Bisector

The point-slope form of a line is given by \(y - y_1 = m (x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope of the line. The midpoint of the line segment OP, which is also a point on the bisector, is \((a/2, a^2/2)\). Plugging these into the formula gives the equation of the bisector as \(y - a^2/2 = (1/a)(x - a/2)\)

Step 4: Find y-intercept of the Perpendicular Bisector

For the y-intercept, set \(x = 0\) in the equation of the line. This yields \(b = y = a^2/2 - a/2\)

Step 5: Compute the Limit of b as P Approaches to O

As \(P \rightarrow O\), this means that \(a \rightarrow 0\). Therefore, we need to calculate \(\lim _ { a \rightarrow 0 } b\). Replacing b with its equation from Step 4 gives \(\lim _ { a \rightarrow 0 } (a^2/2 - a/2)\), which equals -1/2

Key Concepts

Perpendicular Bisector

A perpendicular bisector is a line which cuts another line segment into two equal parts at a right angle. In geometry, it plays a critical role in identifying symmetry and equal distances. When two points form a line segment, the perpendicular bisector is the line that not only divides the segment into equally long parts but does this while maintaining a perpendicular orientation.
For instance, in this exercise, we look at the line segment formed by the origin point, O, and another point, P, on a parabola. The perpendicular bisector of OP has a slope that is the negative reciprocal of the slope of OP. If the slope of OP is \(-a\), then the slope of the perpendicular bisector is \(1/a\), making it crucial for our calculation.

• Visually, it is easy to picture the bisector as a line crossing the midpoint of OP at a 90-degree angle.
• Mathematically, this concept aids in finding relations such as y-intercepts and evaluating various limits related to lines and curves.

Parabola

A parabola is one of the simple yet fundamental shapes in algebra and calculus, defined by a quadratic equation, typically of the form \(y = x^2\).
Parabolas are symmetric around their vertex, which acts as the turning point, and they extend infinitely in both directions along the x-axis. These shapes are characterized by their unique properties, such as having a single axis of symmetry, a U-shaped curve, and a focus and directrix.

• The parabolic point P, \( (a, a^2)\), represents any point on the parabola defined for \(y = x^2\).
• Understanding the geometric properties of a parabola helps us find the position of P on the curve as well as calculate important geometric principles like midpoints and bisectors.

In this problem, the point P moves along the parabola, linking algebraic expressions to geometric interpretations.

Limit Calculation

Limits are a core concept in calculus, representing the value a function approaches as the input approaches some point. In this exercise, finding the limit involves understanding how variables behave as they reach certain boundaries.
Here, we calculate the limit of \( b \) as \( P \) approaches \( O \), which entails \( a \) approaching \( 0 \).

• The expression for \( b \) as shown is \( a^2/2 - a/2\).
• Taking the limit as \( a \rightarrow 0\) simplifies this expression to \(-1/2\).

This kind of calculation is essential in various fields of science and engineering, where determining the behavior of systems as they approach certain constraints or conditions is key.