Problem 33
Question
Give a geometric interpretation of the set $$ A=\left\\{(x, y) \in \mathbf{R}^{2}: \sqrt{x^{2}+y^{2}-4 y+4}<3\right\\} $$
Step-by-Step Solution
Verified Answer
Set A represents the interior of a circle centered at (0, 2) with radius 3.
1Step 1: Simplify the Expression Inside the Square Root
We start with the expression under the square root: \( \sqrt{x^2 + y^2 - 4y + 4} < 3 \). Notice that the expression \( x^2 + y^2 - 4y + 4 \) can be rearranged. Let's complete the square for the \( y \) terms.
2Step 2: Complete the Square for the y Terms
Rewrite \( y^2 - 4y + 4 \) as \( (y-2)^2 \). Thus, we have: \( x^2 + (y-2)^2 < 3^2 \) which simplifies to \( x^2 + (y-2)^2 < 9 \).
3Step 3: Identify the Geometric Shape
The inequality \( x^2 + (y-2)^2 < 9 \) represents all the points \((x, y)\) that are inside a circle centered at (0, 2) with a radius of 3. This means that the set A is the interior of this circle.
Key Concepts
Completing the SquareCircle EquationGeometric Shapes
Completing the Square
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly handy when dealing with circle equations or any quadratic expression. To complete the square, especially for exercises like this, you focus on expressions like \( y^2 - 4y + 4 \). Here's how it works:
* First, take the coefficient of \( y \), which is -4, and divide it by 2, giving you -2. * Next, square this result to get 4. * Therefore, the expression \( y^2 - 4y + 4 \) becomes \((y-2)^2\).
This conversion is useful because it makes the equation more recognizable and easier to manipulate algebraically. Completing the square transforms complex quadratic terms into a format that reveals the geometric nature of equations more clearly, especially useful in geometric interpretations.
* First, take the coefficient of \( y \), which is -4, and divide it by 2, giving you -2. * Next, square this result to get 4. * Therefore, the expression \( y^2 - 4y + 4 \) becomes \((y-2)^2\).
This conversion is useful because it makes the equation more recognizable and easier to manipulate algebraically. Completing the square transforms complex quadratic terms into a format that reveals the geometric nature of equations more clearly, especially useful in geometric interpretations.
Circle Equation
The equation we derived, \( x^2 + (y-2)^2 < 9 \), is a classic representation of a circle in algebraic terms. Understanding circle equations is crucial for visualizing what a set of points represent. Here’s a breakdown:
* The general form for a circle's equation is \( (x-h)^2 + (y-k)^2 = r^2 \), where * \((h, k)\) is the center of the circle, * and \(r\) is the radius.* In our equation, \( (x-0)^2 + (y-2)^2 < 9 \), it tells us that: * The center of the circle is at \((0, 2)\). * The circle has a radius of 3 (since \( r^2 = 9 \), meaning \( r = \sqrt{9} = 3 \)).
This form allows us to easily identify the circle's center and radius, simplifying the process of graphing and solving geometric problems involving circles.
* The general form for a circle's equation is \( (x-h)^2 + (y-k)^2 = r^2 \), where * \((h, k)\) is the center of the circle, * and \(r\) is the radius.* In our equation, \( (x-0)^2 + (y-2)^2 < 9 \), it tells us that: * The center of the circle is at \((0, 2)\). * The circle has a radius of 3 (since \( r^2 = 9 \), meaning \( r = \sqrt{9} = 3 \)).
This form allows us to easily identify the circle's center and radius, simplifying the process of graphing and solving geometric problems involving circles.
Geometric Shapes
Geometric shapes form the basis of many algebraic interpretations, especially when interpreting sets of points like in this exercise. Here, the solution tells us we’re dealing with the interior of a circle. But how does a concept like this translate into understanding geometric shapes?
* A circle is one of the simplest 2-D geometric shapes, defined by all the points that are equidistant from a center point in a plane.* When we have an inequality, \( (x-0)^2 + (y-2)^2 < 9 \), rather than an equation, it tells us about the area _inside_ the circle. * This interpretation means that the set \( A \) contains every point \( (x, y) \) that lies within a circle having 3 as its maximum distance (radius) from the center \((0, 2)\).
Understanding these concepts of circles and other geometric shapes through equations not only aids in visualization but also in solving practical geometric problems. Geometric shapes in math often translate complex algebra into visually understandable and meaningful interpretations.
* A circle is one of the simplest 2-D geometric shapes, defined by all the points that are equidistant from a center point in a plane.* When we have an inequality, \( (x-0)^2 + (y-2)^2 < 9 \), rather than an equation, it tells us about the area _inside_ the circle. * This interpretation means that the set \( A \) contains every point \( (x, y) \) that lies within a circle having 3 as its maximum distance (radius) from the center \((0, 2)\).
Understanding these concepts of circles and other geometric shapes through equations not only aids in visualization but also in solving practical geometric problems. Geometric shapes in math often translate complex algebra into visually understandable and meaningful interpretations.
Other exercises in this chapter
Problem 33
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