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 as the distance between centers (13) exceeds the sum of their radii (12).

Step 1: Rearrange circle equations

The given circle equations need to match the standard circle equation form: \[(x-h)^2 + (y-k)^2 = r^2\] where \(h,k\) is the center, and \r\ is the radius of the circle. Begin by rearranging each equation to reveal the center and radius.

Step 2: Complete the square for Circle 1

The first equation is \(x^2 + y^2 - 4x - 2y - 11 = 0\). Complete the square separately for \(x\) and \(y\):- For \(x\): \((x^2 - 4x)\) becomes \((x-2)^2 - 4\)- For \(y\): \((y^2 - 2y)\) becomes \((y-1)^2 - 1\)The equation becomes \((x-2)^2 + (y-1)^2 = 16\), giving center (2, 1) and radius 4.

Step 3: Complete the square for Circle 2

The second equation is \(x^2 + y^2 + 20x - 12y + 72 = 0\). Complete the square separately for \(x\) and \(y\):- For \(x\): \((x^2 + 20x)\) becomes \((x+10)^2 - 100\)- For \(y\): \((y^2 - 12y)\) becomes \((y-6)^2 - 36\)The equation becomes \((x+10)^2 + (y-6)^2 = 64\), giving center (-10, 6) and radius 8.

Step 4: Calculate distance between centers

Using the formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\):\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute the centers (2, 1) and (-10, 6):\[ \sqrt{(-10 - 2)^2 + (6 - 1)^2} = \sqrt{(-12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

Step 5: Compare distance to sum of radii

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

Key Concepts

Equation of a Circle

The equation of a circle is a fundamental concept in circle geometry. It is expressed in the standard form as \((x-h)^2 + (y-k)^2 = r^2\). In this equation, \((h, k)\) represents the center of the circle, and \(r\) is the radius. Understanding this formula is essential when working with problems involving circles, as it provides a direct way to identify a circle's center and size.

To convert from a general quadratic equation form, like \(x^2 + y^2 - 4x - 2y - 11 = 0\), into the standard circle equation form, completing the square is used. This involves manipulating the equation to create a perfect square trinomial.

Knowing the equation in its standard form helps in analyzing and visualizing the circle, leading to a more intuitive understanding of its geometry.

Completing the Square

Completing the square is a key technique used to transform quadratic equations into a recognizable form. It's especially useful in circle geometry to convert general quadratic equations into the standard equation of a circle.

Here’s how it works:

• For a quadratic expression in the form \(x^2 + bx\), we add and subtract \((\frac{b}{2})^2\) inside the equation. This rearranges it into a perfect square trinomial: \((x + \frac{b}{2})^2 - (\frac{b}{2})^2\).
• This method is applied separately to both \(x\)-terms and \(y\)-terms in a circle equation.

Once done, the circle's equation becomes easier to interpret, allowing you to read the center and radius directly from the equation.

Center and Radius of a Circle

Identifying the center and radius of a circle is crucial in understanding its geometry. In the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), the center is the point \((h, k)\), and the radius is \(r\).

To find these elements from a given equation:

• First, complete the square for both \(x\) and \(y\) components.
• Convert the expression into its standard form, which will expose the \(h\), \(k\), and \(r\) values.

Understanding these components helps in solving geometric problems such as finding intersections, tangents, and distances efficiently.

Distance Between Two Points

The distance between two points is a measure of the space separating them in the coordinate plane. This concept is vital when determining relationships between geometric figures, such as circles in this case.

The formula used to compute this distance between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

To ensure clear understanding:

• Calculate the difference between the respective coordinates: \(x_2 - x_1\) and \(y_2 - y_1\).
• Square these differences, add the results, then take the square root.

This formula helps ascertain whether circles intersect by comparing this distance to the sum of their radii.