Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

.\( Find the distance between the points on the circle \)x^{2}+y^{2}=13\( with the \)x\( -coordinates \)-2\( and \)2 .$ How many such distances are there?

Step-by-Step Solution

Answer

There are two distinct distances: 4 and 6.

Step 1: Find the y-coordinates of the points

Given the equation of the circle \( x^2 + y^2 = 13 \), for \( x = -2 \), substitute into the equation: \( (-2)^2 + y^2 = 13 \). Simplifying gives \( 4 + y^2 = 13 \)\( \Rightarrow y^2 = 9 \). Thus, \( y = 3 \) or \( y = -3 \). Similarly, for \( x = 2 \), \( 2^2 + y^2 = 13 \). Again, \( y^2 = 9 \) giving \( y = 3 \) or \( y = -3 \). Thus, the points are \((-2, 3)\), \((-2, -3)\), \((2, 3)\), and \((2, -3)\).

Step 2: Calculate the distances between points with same x-coordinates

The distance between \((-2, 3)\) and \((-2, -3)\) is \(|3 - (-3)| = |3 + 3| = 6\), and similarly, the distance between \((2, 3)\) and \((2, -3)\) is \(|3 - (-3)| = 6\).

Step 3: Calculate the distances between points with opposite x-coordinates

Calculate the distance between \((-2, 3)\) and \((2, 3)\) using the distance formula: \( \sqrt{(2 + 2)^2 + (3 - 3)^2} = \sqrt{4^2} = 4 \). The same logic applies to \((-2, -3)\) and \((2, -3)\), which also gives a distance of \(4\).

Key Concepts

Circle Equation

A circle equation represents all the points that lie on a circle in a coordinate system. The standard form of the circle equation is \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle. Circle equations are crucial in coordinate geometry because they define a precise boundary that every point on the perimeter satisfies. For instance, in the problem we’re exploring, the equation \( x^2 + y^2 = 13 \) implies a circle with a radius of \( \sqrt{13} \). Every point \((x, y)\) that satisfies this equation lies on the circle. The center of this circle is at the origin \((0, 0)\) of the coordinate plane. Understanding this equation allows us to visualize the circle's placement and proportion in the plane.
Remember that manipulating a circle equation by changing the values of \( x \) or \( y \) allows you to solve for the other variable, which will tell you exactly where the points lie on this circular path. This is the grounding element used in problem-solving when dealing with points on a circle.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In this system, you can use coordinates to express geometric figures and solve geometrical problems.
It's a useful tool because it allows for algebraic manipulation to define and analyze shapes, like circles and lines, on a graph. You work with points defined by pairs of numbers \((x, y)\). Each point on a plane can be graphed and calculated geometrically.
In our example, you find the points that lie on a circle using their \( x \)-coordinates, further demystifying their placement in relation to each other. This problem-solving geometry technique makes it possible to derive meaningful information, like distances between points, by treating the elements of the figure as algebraic equations.

Distance Formula

The distance formula is a fundamental tool in coordinate geometry used to determine the distance between two points on a plane. The formula is derived from the Pythagorean theorem and is given by:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In simpler terms, it calculates the hypotenuse length between two points based on the differences in their x and y coordinates. This formula allows you to comprehend the spatial difference between points easily and accurately.
In our exercise, the distance formula is applied to find the length between points on a circle, utilizing the set coordinates of each point. For example, to find the distance between \((-2, 3)\) and \((2, 3)\), substitute the coordinates into the formula to yield \( \sqrt{(2 - (-2))^2 + (3 - 3)^2} = \sqrt{4^2} = 4 \).
This methodology streamlines the process of finding distances, ensuring that even if the points are complex or located on different sections of the plane, you can accurately establish their separation.

Problem Solving in Geometry

Problem-solving in geometry involves using mathematical principles, formulas, and strategies to solve questions concerning shapes, sizes, and other properties of space. In geometry, understanding the relationships and properties between figures and measurements is vital.
This particular exercise demonstrates key problem-solving skills by:

• Interpreting circle equations to identify pairs of coordinates.
• Applying the distance formula effectively to calculate distances between points.
• Evaluating multiple potential solutions, such as different possible distances between pairs of points on a circle, to confirm results.

Solving geometrical problems demands analytical thinking and a systematic approach, particularly when dealing with sophisticated structures such as circles. By using tried and tested formulas, translating complex shapes into manageable algebraic tasks becomes trivial. This exercise showcases all these aspects proficiently, allowing for a deeper grasp of geometry as a logical and structured discipline.