Problem 31
Question
write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(0,0), r=7 $$
Step-by-Step Solution
Verified Answer
The standard form of the equation of the circle with the given center and radius is \( x^2 + y^2 = 49 \).
1Step 1: Identify the values from the problem
Identify the coordinates of the center and the radius from the given problem. Here, the center of the circle is at (0,0) and the radius of the circle is 7.
2Step 2: Substitute the values into the standard form equation
Substitute the values of the coordinates of the center \((h, k) = (0,0)\) and the radius \(r = 7\) into the standard equation of the circle: \( (x-h)^2 + (y-k)^2 = r^2 \). This gives the equation as \( (x-0)^2 + (y-0)^2 = 7^2 \)
3Step 3: Simplify the equation
Simplify the equation to its simplest form. This results in the equation of the circle as \( x^2 + y^2 = 49 \).
Key Concepts
Circle GeometryCoordinate GeometryAlgebraic Equations
Circle Geometry
Understanding circle geometry is essential for solving problems related to the shapes and sizes of circles. A circle is defined as the set of all points in a plane that are at a fixed distance, known as the radius, from a given point called the center.
When dealing with circles in a coordinate plane, the standard form equation is vital. It represents every point on the circle as an ordered pair \( (x, y) \) satisfying the equation \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) are the coordinates of the center, and \( r \) is the radius.
For a circle centered at the origin \( (0,0) \) with radius \( r \) the equation simplifies, since \( h \) and \( k \) both equal 0. Therefore, the equation \( x^2 + y^2 = r^2 \) represents a circle with a center at the origin. The radius determines the size of the circle and can be plotted by measuring distance from the center to any point on the circumference of the circle.
When dealing with circles in a coordinate plane, the standard form equation is vital. It represents every point on the circle as an ordered pair \( (x, y) \) satisfying the equation \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) are the coordinates of the center, and \( r \) is the radius.
For a circle centered at the origin \( (0,0) \) with radius \( r \) the equation simplifies, since \( h \) and \( k \) both equal 0. Therefore, the equation \( x^2 + y^2 = r^2 \) represents a circle with a center at the origin. The radius determines the size of the circle and can be plotted by measuring distance from the center to any point on the circumference of the circle.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to analyze geometrical shapes using a coordinate system.
In this approach, points on a plane are identified by their distances from two intersecting, perpendicular lines called axes, labeled as the \( x \) and \( y \) axes. Positions or coordinates are then expressed as ordered pairs \( (x, y) \). The concept is crucial when working with equations of geometric figures like lines, rectangles, and, as in this instance, circles.
The power of coordinate geometry lies in its ability to convert geometric problems into algebraic ones, which can make them easier to solve systematically. For example, the standard form equation of a circle \( (x-0)^2 + (y-0)^2 = 7^2 \) describes a perfect circle in a coordinate plane where every \( (x, y) \) that satisfies the equation is a point on the circle.
In this approach, points on a plane are identified by their distances from two intersecting, perpendicular lines called axes, labeled as the \( x \) and \( y \) axes. Positions or coordinates are then expressed as ordered pairs \( (x, y) \). The concept is crucial when working with equations of geometric figures like lines, rectangles, and, as in this instance, circles.
The power of coordinate geometry lies in its ability to convert geometric problems into algebraic ones, which can make them easier to solve systematically. For example, the standard form equation of a circle \( (x-0)^2 + (y-0)^2 = 7^2 \) describes a perfect circle in a coordinate plane where every \( (x, y) \) that satisfies the equation is a point on the circle.
Algebraic Equations
Algebraic equations are mathematical statements that express the equality between two algebraic expressions. These equations are composed of variables, constants, and arithmetic operations. They serve as a foundation for solving a wide range of problems in mathematics, including those involving shapes and their properties.
In the context of circle geometry, the algebraic equation takes the form of the standard equation of a circle. It is a way of expressing all the points that lie on the circumference of a circle via an equation. By manipulating these equations, we can find centers, radii, and plot the circle on a coordinate grid.
The equation \( x^2 + y^2 = 49 \) is an example of an algebraic equation that represents a circle with radius 7 centered at the origin. It showcases how algebra can be used to describe a geometric shape, allowing for problem-solving using algebraic techniques such as squaring numbers, simplifying expressions, and solving for variables.
In the context of circle geometry, the algebraic equation takes the form of the standard equation of a circle. It is a way of expressing all the points that lie on the circumference of a circle via an equation. By manipulating these equations, we can find centers, radii, and plot the circle on a coordinate grid.
The equation \( x^2 + y^2 = 49 \) is an example of an algebraic equation that represents a circle with radius 7 centered at the origin. It showcases how algebra can be used to describe a geometric shape, allowing for problem-solving using algebraic techniques such as squaring numbers, simplifying expressions, and solving for variables.
Other exercises in this chapter
Problem 30
Find the domain of each function. $$f(x)=\frac{7 x+2}{x^{3}-2 x^{2}-9 x+18}$$
View solution Problem 30
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-2,-4)\) and \((1,-1)\)
View solution Problem 31
Evaluate each function at the given values of the independent variable and simplify. $$ h(x)=x^{4}-x^{2}+1 $$ a. \(h(2)\) b. \(h(-1)\) c. \(h(-x)\) d. \(h(3 a)\
View solution Problem 31
In Exercises \(31-50,\) find \(f+g, f-g,\) fg, and \(\frac{f}{g} .\) Determine the domain for each function. $$f(x)=2 x+3, g(x)=x-1$$
View solution