Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the two circles \(x^{2}+y^{2}-4 x-2 y-11=0\) and \(x^{2}+y^{2}+20 x-12 y+72=0\) do not intersect. Hint: Find the distance between their centers.

Step-by-Step Solution

Answer

The circles do not intersect because the distance between centers (13) is greater than the sum of their radii (12).

Step 1: Identify the Circle Equations

The given equations for the circles are:1. \( C_1: x^2 + y^2 - 4x - 2y - 11 = 0 \)2. \( C_2: x^2 + y^2 + 20x - 12y + 72 = 0 \).We need to find the centers and radii of these circles.

Step 2: Find Center and Radius of the First Circle

For the first circle equation \( x^2 + y^2 - 4x - 2y - 11 = 0 \), we complete the square:1. Rearrange terms: \( (x^2 - 4x) + (y^2 - 2y) = 11 \).2. Complete the square: - \( x^2 - 4x = (x - 2)^2 - 4 \) - \( y^2 - 2y = (y - 1)^2 - 1 \)3. Substitute back: \( (x - 2)^2 + (y - 1)^2 = 16 \).Thus, the center of the first circle, \( C_1 \), is \((2, 1)\) and the radius \( r_1 \) is 4.

Step 3: Find Center and Radius of the Second Circle

For the second circle equation \( x^2 + y^2 + 20x - 12y + 72 = 0 \), complete the square:1. Rearrange: \( (x^2 + 20x) + (y^2 - 12y) = -72 \).2. Complete the square: - \( x^2 + 20x = (x + 10)^2 - 100 \) - \( y^2 - 12y = (y - 6)^2 - 36 \)3. Substitute back: \( (x + 10)^2 + (y - 6)^2 = 64 \).Thus, the center of the second circle, \( C_2 \), is \((-10, 6)\) and the radius \( r_2 \) is 8.

Step 4: Calculate Distance Between the Centers

The distance between the centers \((2, 1)\) and \((-10, 6)\) can be found using the distance formula:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substituting the coordinates:\[ d = \sqrt{((-10) - 2)^2 + (6 - 1)^2} = \sqrt{(-12)^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \].

Step 5: Compare Sum of Radii with Distance Between Centers

Sum of the radii of the circles is \( r_1 + r_2 = 4 + 8 = 12 \).The distance between the centers is 13. Since the distance between the centers is greater than the sum of the radii, the circles do not intersect.

Key Concepts

Circle Equations

In geometry, understanding circle equations is crucial when analyzing their properties. A circle's standard equation form is \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) represents the circle's center coordinates and \( r \) represents its radius. Circle equations can also appear in a general quadratic form like \( x^2 + y^2 + Dx + Ey + F = 0 \). In such cases, to identify the circle's center and radius, we rearrange and complete the square. This transformation leads to a clearer depiction of the circle through its standard form, unveiling its geometric characteristics. The given problem involves two such equations, where extracting the circle's parameters allows us to further analyze their geometric relationship.

Circle Intersection

Determining whether two circles intersect involves comparing the distance between their centers with the sum of their radii. Intersection can occur in three scenarios:

• If the distance equals the sum of the radii, the circles touch externally (tangent intersection).
• If the distance is less than the sum but more than the difference of the radii, they intersect at two points.
• If the distance is greater than the sum, the circles do not intersect.

The exercise here concludes with a non-intersection because the distance between the centers is greater than the sum of the radii. Insight into their interaction helps visualize their spatial arrangement and determine how they relate geometrically in a plane.

Distance Formula

The distance formula is a key tool for calculating the distance between two points in a plane, derived from the Pythagorean theorem. The formula is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). It considers the differences in the x and y coordinates of two points \( (x_1, y_1) \) and \( (x_2, y_2) \). In this context, we use it to find the distance between the centers of the two circles. This computation determines whether the circles intersect. The resulting distance serves as a basis for comparison against other circle parameters, critical for concluding about their intersection properties.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. In circle equations, it's applied to convert the general form \( x^2 + y^2 + Dx + Ey + F = 0 \) into the standard circle form. The method involves:

• Rearranging the terms related to \( x \) and \( y \).
• Adding and subtracting the square of half the coefficients of \( x \) and \( y \).

Completing the square provides the center and radius of the circle, pivotal for analyzing their positions. Mastery of this technique is essential for resolving circle-related problems efficiently as showcased in solving the provided exercise.