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 is one unique distance between these points.

Step 1: Understand the Circle Equation

The equation of the circle is given as \( x^2 + y^2 = 13 \). This represents a circle centered at the origin \((0,0)\) with a radius of \(\sqrt{13}\).

Step 2: Identify Possible Y-Coordinates

We are given two points on the circle with \(x\)-coordinates \(-2\) and \(2\). To find the corresponding \(y\)-coordinates, substitute each \(x\) value into the circle's equation. Start with \(x = -2\): \((-2)^2 + y^2 = 13\), which simplifies to \(4 + y^2 = 13\), hence \(y^2 = 9\). Therefore, \(y = 3\) or \(y = -3\). Repeat the same for \(x = 2\): \(2^2 + y^2 = 13\), leading to the same conclusion, \(y = 3\) or \(y = -3\).

Step 3: Calculate the Distance Between Points

There are four possible points: \((-2,3)\), \((-2,-3)\), \((2,3)\), and \((2,-3)\). Calculate the distance between each pair with the same \(x\)-coordinate: \((-2,3)\) and \((2,3)\) using the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). The distance is \(d = \sqrt{(2 - (-2))^2} = \sqrt{4^2} = 4\). The same calculation applies to \((-2,-3)\) and \((2,-3)\), also giving a distance of 4. This makes two possible distances: from \((-2,3)\) to \((2,3)\) and from \((-2,-3)\) to \((2,-3)\).

Step 4: Count Unique Distances

Since both calculations lead to the same length of \(4\), there is only one unique distance between the points with \(x\)-coordinates \(-2\) and \(2\) on the circle.

Key Concepts

Distance Formula

The distance formula is a handy tool in coordinate geometry. It helps us find the distance between two points on a plane, using their coordinates. The formula is derived from the Pythagorean theorem. In its basic form, it appears as \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).
This formula calculates the length of the line segment connecting the points \((x_1, y_1)\) and \((x_2, y_2)\).
When using the distance formula:

• Identify the coordinates of the two points.
• Subtract the x-coordinates and the y-coordinates separately.
• Square both differences.
• Add the squares together and then take the square root of the sum.

In our exercise, this formula was used to determine the distance between the points with x-coordinates of -2 and 2. Each resulting calculation led to a distance of 4, showcasing how the distance formula simplifies finding lengths between points on a plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that describes the relationships of geometric figures using coordinate systems.
This field offers a bridge between algebra and geometry.
In applying coordinate geometry:

• Points are represented as coordinates \((x, y)\) on the Cartesian plane.
• Shapes such as lines, circles, and polygons can be described with equations.
• Geometric problems become algebra problems.

For our specific exercise, we used the coordinate approach to locate points on a circle defined by the equation \( x^2 + y^2 = 13 \). By substituting x-values provided, we found corresponding y-values, thereby pinpointing exact locations on the circle where distances could be measured.

Circle Equation

A circle's equation provides a mathematical representation of a circle in the coordinate plane. One common form is \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle, and the center is at the origin \((0, 0)\).
To work with circle equations, keep in mind:

• The left-hand side expresses a set of positions (x, y) relative to the circle's center.
• The right-hand side equals the square of the circle's radius.
• Any point \((x, y)\) satisfying the equation lies on the circle's boundary.

In the exercise at hand, the circle is given by \( x^2 + y^2 = 13 \).
This indicates a circle at the origin with a radius of \( \sqrt{13} \).
The exercise involved substituting specific x-values into the circle equation to find corresponding y-values. This process determines the specific points on the circle to be analyzed.