Problem 96 - Algebra and Trigonometry

Question

(a) Find the radius of each circle in the pair and the distance between their centers; then use this information to determine whether the circles intersect. $$\begin{aligned} \text { (i) }(x-2)^{2}+(y-1)^{2} &=9 \\\\(x-6)^{2}+(y-4)^{2} &=16 \\ \text { (ii) } x^{2}+(y-2)^{2}=4 & \\\\(x-5)^{2}+(y-14)^{2} &=9 \\\ \text { (iii) }(x-3)^{2}+(y+1)^{2} &=1 \\\\(x-2)^{2}+(y-2)^{2} &=25 \end{aligned}$$ (b) How can you tell, just by knowing the radii of two circles and the distance between their centers, whether the circles intersect? Write a short paragraph describing how you would decide this, and draw graphs to illustrate your answer.

Step-by-Step Solution

Answer

(i) Intersect; (ii) Do not intersect; (iii) Intersect. Circles intersect if the center distance lies between the sum and the absolute difference of radii.

Step 1: Identify the center and radius of each circle

For each given circle equation in standard form \((x-h)^2 + (y-k)^2 = r^2\), identify the center as \((h, k)\) and the radius as \(r\).

Step 2: Find the circle information for Case (i)

Circle 1: From equation \((x-2)^2+(y-1)^2=9\).- Center: \((2, 1)\)- Radius: \(\sqrt{9} = 3\)Circle 2: From equation \((x-6)^2+(y-4)^2=16\).- Center: \((6, 4)\)- Radius: \(\sqrt{16} = 4\).Distance between centers: \(\sqrt{(6-2)^2 + (4-1)^2} = \sqrt{9 + 16} = 5\).

Step 3: Determine intersection for Case (i)

For two circles to intersect, the distance between their centers \(d\) should be less than or equal to the sum of their radii but greater than or equal to the absolute difference of their radii.- \(d = 5\)- Sum of radii: \(3 + 4 = 7\)- Difference of radii: \(|3 - 4| = 1\)Since \(1 \leq 5 \leq 7\), the circles intersect.

Step 4: Find the circle information for Case (ii)

Circle 1: From equation \(x^2+(y-2)^2=4\).- Center: \((0, 2)\)- Radius: \(\sqrt{4} = 2\).Circle 2: From equation \((x-5)^2+(y-14)^2=9\).- Center: \((5, 14)\)- Radius: \(\sqrt{9} = 3\).Distance between centers: \(\sqrt{(5-0)^2 + (14-2)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\).

Step 5: Determine intersection for Case (ii)

The sum of the radii is \(2 + 3 = 5\), and the difference is \(|2 - 3| = 1\).The distance between centers is \(13\), which is greater than the sum \(5\), so the circles do not intersect.

Step 6: Find the circle information for Case (iii)

Circle 1: From equation \((x-3)^2+(y+1)^2=1\).- Center: \((3, -1)\)- Radius: \(\sqrt{1} = 1\).Circle 2: From equation \((x-2)^2+(y-2)^2=25\).- Center: \((2, 2)\)- Radius: \(\sqrt{25} = 5\).Distance between centers: \(\sqrt{(3-2)^2 + (-1-2)^2} = \sqrt{1 + 9} = \sqrt{10}\).

Step 7: Determine intersection for Case (iii)

The sum of the radii is \(1 + 5 = 6\), and the difference is \(|1 - 5| = 4\).The distance \(\sqrt{10}\) or approximately \(3.16\) is less than the sum \(6\) and greater than the difference \(4\), so the circles intersect.

Step 8: Determine intersection rule description

Two circles intersect if the distance \(d\) between their centers is such that \(|r_1 - r_2| \leq d \leq r_1 + r_2\). If \(d < |r_1 - r_2|\) or \(d > r_1 + r_2\), the circles do not intersect. On a graph, this means the circles overlap or just touch if this condition is met.

Key Concepts

Circle Equations

Circle equations in their standard form are expressed as \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius. These equations provide the essential information needed to graph a circle and therefore to analyze problems involving circles.

• The \(h\) and \(k\) are the coordinates of the center, which tells us where the circle is positioned on the Cartesian plane.
• The \(r^2\) is the square of the radius. Taking the square root of this value gives us the radius, telling us how large the circle is.

Understanding circle equations is crucial in geometry especially when dealing with problems involving intersections or distances between two circles. Each element of the equation plays a specific role in defining the properties of the circle. By comparing these equations, one can start solving intersection problems by determining the relative position and size of each circle.

Distance Between Centers

The distance between the centers of two circles is an important metric in determining their relationship to one another. It is calculated using the distance formula:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula computes the straight-line distance between two points in the plane, known as the Euclidean distance. To find this:

• Identify the centers \((x_1, y_1)\) and \((x_2, y_2)\) from each circle's equation.
• Substitute the coordinates into the distance formula to find \(d\).

Knowing the distance between the centers helps in visualizing how the two circles are positioned relative to each other, and it also aids in determining whether the circles overlap, touch, or are totally separate.

Radii Comparison

The comparison of radii in conjunction with the distance between circle centers is key to understanding circle intersections. To evaluate whether two circles intersect, compare the sum and the absolute difference of their radii to their center distance:

• If the distance is between the absolute difference and the sum of the radii, i.e., \(|r_1 - r_2| \leq d \leq r_1 + r_2\), the circles intersect.
• If \(d > r_1 + r_2\), the circles are separate and do not touch.
• If \(d < |r_1 - r_2|\), one circle is completely inside the other without touching.

This method of radii comparison helps quickly determine intersection possibilities without needing to plot the circles. It uses mathematical relationships to logically deduce the spatial arrangement of the circles.