Problem 15 - Basic Engineering Mathematics

Question

Determine (a) the radius, and (b) the co-ordinates of the centre of the circle given by the equation: $$ x^{2}+y^{2}+8 x-2 y+8=0 $$

Step-by-Step Solution

Answer

Center: (-4, 1), Radius: 3.

Step 1: Identify the Standard Form

The equation of a circle is generally given in the form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) are the coordinates of the center and \(r\) is the radius. First, we need to rewrite the given equation in this form.

Step 2: Rearrange the Terms

Start by rearranging the given equation \(x^2 + y^2 + 8x - 2y + 8 = 0\) to group \(x\) and \(y\) terms: \(x^2 + 8x + y^2 - 2y = -8\).

Step 3: Complete the Square for x

To complete the square for the \(x\) terms \(x^2 + 8x\), take the coefficient of \(x\), which is 8, divide it by 2 to get 4, and square it to get 16. Add and subtract 16 to rewrite as \((x+4)^2 - 16\).

Step 4: Complete the Square for y

Similarly, complete the square for the \(y\) terms \(y^2 - 2y\). The coefficient of \(y\) is -2, divide by 2 to get -1, square it to get 1. Add and subtract 1: \((y-1)^2 - 1\).

Step 5: Substitute and Simplify

Substitute the completed squares into the equation and solve for constants: \((x+4)^2 - 16 + (y-1)^2 - 1 = -8\). Simplify to get \((x+4)^2 + (y-1)^2 = 9\).

Step 6: Identify Center and Radius

From \((x+4)^2 + (y-1)^2 = 3^2\), it's clear that the center of the circle is \((-4, 1)\) and the radius is \(r = 3\).

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a more workable form, particularly useful for circle equations. This transformation helps in identifying the center and the radius of a circle. To complete the square in an equation involving a circle, focus on grouping the terms involving the same variable together.

In our problem, we start by finding the square for the terms dealing with variable \( x \), which are \( x^2 + 8x \). First, take the coefficient of \( x \), which is 8, and divide it by 2 to get 4. Square this result, leading to 16. Thus, the expression \( x^2 + 8x \) can be rewritten as \((x+4)^2 - 16\).

Repeat this process for the \( y \) variable terms, \( y^2 - 2y \). Take the coefficient, -2, divide by 2 to get -1, then square it to get 1. Thus, \( y^2 - 2y \) is rewritten as \((y-1)^2 - 1\). These transformations show how the circle's terms can be transformed into a perfect square form, revealing the circle's geometric properties.

Circle Geometry

Circle geometry is the study of properties and definitions associated with circles in mathematics. A circle is a set of all points in a plane that are equidistant from a given point, called the center. The equation of a circle in standard form makes it simple to identify these critical properties. For instance, given a transformed circle equation like \((x - h)^2 + (y - k)^2 = r^2\), the coordinate \( (h, k) \) represents the center of the circle and \( r \) represents the radius.

In our problem, after completing the square, the equation reshapes into \((x+4)^2 + (y-1)^2 = 9\). Translated into the standard form, it indicates that the circle's center is at \((-4, 1)\) and the radius is 3, derived from \( r^2 = 9\), thus \( r = 3\).

Understanding these geometric components is essential for interpreting and solving questions related to circles as they help to visually and numerically capture the essence of a circle's properties.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to study geometry using a coordinate system. It is exceptionally useful for solving geometric problems by converting them into algebraic equations. Circles provide perfect examples where coordinate geometry comes into play. By defining a circle's equation, you can mathematically locate its center or determine the radius using algebraic methods.

In our current scenario, using coordinate geometry helps translate the complex structure of a circle into an understandable form. Transforming the given equation into a standard form involves using coordinate geometry concepts. By finding the center \((-4, 1)\) and radius \(3\) from the standard circle equation \((x+4)^2 + (y-1)^2 = 9\), we can see how coordinate geometry simplifies locating a circle's key features in a plane.

Coordinate geometry also offers tools for verifying solutions and visualizing geometric positions, aiding in plotting circles on a graphical plane or solving intersection problems with lines or other circles.