Problem 70 - Contemporary Precalculus

Question

Do the circles with the following equations intersect? $$(x-3)^{2}+(y+2)^{2}=25 \quad \text{and} \quad(x+3)^{2}+(y-2)^{2}=4$$ [Hint: Consider the radii and the distance between the centers. \(]\)

Step-by-Step Solution

Answer

Briefly explain your answer. Answer: Yes, the circles intersect. This is because the sum of their radii, 7, is greater than the distance between their centers, which is √52.

Step 1: Analyze the equations and identify the centers and radii

Given the equations of both circles: $$(x-3)^2 +(y+2)^2=25 \quad \text{and} \quad (x+3)^2 +(y-2)^2=4$$ Their centers can be identified as \((3, -2)\) and \((-3, 2)\) respectively. In the same way we can identify their radius by the square root of the constants 25 and 4 So, radius of circle 1: \(r_1 = \sqrt{25} = 5\) radius of circle 2: \(r_2 = \sqrt{4} = 2\)

Step 2: Calculate the distance between the centers

To find the distance between the centers of the circles, we will use the distance formula: $$d = \sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}$$ Using the coordinates of the centers, the distance between the two circles is: $$d = \sqrt{(-3-3)^2 +(2 - (-2))^2} = \sqrt{(-6)^2 + (4)^2} = \sqrt{52}$$

Step 3: Determine if the circles intersect

To determine if the circles intersect, we need to compare the sum of their radii with the distance between their centers. There are three possible cases: 1. If \(r_1 + r_2 > d\), the circles intersect. 2. If \(r_1 + r_2 = d\), the circles are tangent (touch at exactly one point). 3. If \(r_1 + r_2 < d\), then the circles do not intersect. Now, we have \(r_1 = 5\), \(r_2 = 2\) and \(d = \sqrt{52}\) Evaluating the sum of radii: \(r_1 + r_2 = 5 + 2 = 7\) We can see that \(7 > \sqrt{52}\), which means the given circles intersect.

Key Concepts

Distance Formula

The distance formula is a fundamental concept in geometry, particularly when working with points and lines on a coordinate plane. It is derived from the Pythagorean Theorem and allows us to calculate the distance between any two points with known coordinates. The formula states that if you have two points, point 1 with coordinates \( (x_1, y_1) \) and point 2 with coordinates \( (x_2, y_2) \) then the distance \(d\) between these points is given by:
\[ d = \sqrt{(x_2-x_1)^2 +(y_2-y_1)^2} \]
Imagine walking from one point to another; the distance formula calculates the shortest path, like a straight line you would draw between these two points on paper. The formula is immensely useful in many areas of mathematics, including our problem about circle intersections, as it helps to determine how far apart the circle centers are.

Circle Equations

A circle's equation on a coordinate plane provides a vital connection between algebra and geometry. The standard form of a circle's equation centered at point \( (h, k) \) with radius \( r \) is given by:
\[ (x-h)^2 + (y-k)^2 = r^2 \]
In the case of circle intersections, we have two such equations, each representing a unique circle. By faithfully representing a circle in algebraic form, these equations allow us to use calculus to solve geometrical problems. Unpacking the formula reveals the center of the circle at \( (h, k) \) and its radius as the square root of the value on the right side of the equation. Analyzing such equations is the first step towards solving many geometric problems involving circles, such as finding points of intersection, understanding tangency, or even computing areas and perimeters.

Radii of Circles

The radii of circles are a measure of their size – the length from the center to any point on the circumference. It's what defines the thickness of pizza crust when you look at it sideways, or how wide a roundabout is as you drive around it. In the context of our problem regarding circle intersections, radii play a critical role. Generally speaking, the larger the radius, the larger the circle. Knowing the radii of both circles involved is essential to determine their relationship to one another: whether they intersect, touch at a single point (tangent), or are completely separate.

It's important to note that the singular form of radii is radius. On a coordinate plane, if you increase the radius, the circle swells outward uniformly from its center. Understanding the radii is also crucial for various applications beyond geometry, such as physics (think about the orbits of planets) or engineering (like gears and pulleys).