91Ó°ÊÓ

When we talk about the equation of a circle, we're diving into one of the fundamental aspects of geometry. At its core, the standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where

• \((h, k)\) represents the center of the circle,
• \(r\) is the radius.

For an equation to describe a circle, the expression should fit this format. In this case, our original equation was \( x^2 + y^2 + 4x - 6y + 13 = 0 \). Notice, initially, it doesn't appear in the standard form.Ultimately, our goal is to convert this general form into the standard form. Taking into account both the \(x\) and \(y\) components, this transformation helps in identifying the circle's key characteristics: its center and radius. Once transformed, it becomes easier to interpret and plot the circle on a graph.

Completing the square is a crucial algebraic technique often used to transform quadratic expressions into a form that's easier to interpret. In the context of circles, it helps us reframe the equation into a standard form. Let's break down how it's done:For the \(x\)-terms in our equation, we see \(x^2 + 4x\). Here's how you complete the square:

• Take the coefficient of \(x\) (which is 4),
• Halve it (resulting in 2),
• Then square it (getting 4).

Add and subtract this square inside the expression to balance it. This process is mirrored for the \(y\)-terms: For \(y^2 - 6y\), take \(-6\), halve to get \(-3\), and square it for 9.These steps translate the terms into perfect square trinomials. They look like this: \((x + 2)^2 - 4\) and \((y - 3)^2 - 9\). Combining them into the circle equation adds clarity and simplifies the task of identifying the circle's characteristics.

Graph translation involves shifting the entire graph of an equation to a new position without altering its shape. Once we have the circle's equation in a perfect square form, identifying translations becomes a breeze.In the equation \((x + 2)^2 + (y - 3)^2 = r^2\), notice that the transformations \(x + 2\) and \(y - 3\) suggest shifts.

• The \(+2\) with \(x\) means the graph shifts left by 2 units.
• The \(-3\) with \(y\) indicates a shift upwards by 3 units.

Such transformations adjust the circle’s center from the origin to \((-2, 3)\). When plotted, the original axes (\(x\) and \(y\)) can be imagined as new axes (\(X = x + 2\) and \(Y = y - 3\)). For our specific example, since the radius is \(0\), this translates into a single point; namely the center of what would have been a circle, marked at \((-2, 3)\) on the graph.