Question

Write the equation of the given ellipse in standard form. . \(x^{2}+y^{2}-4 x-4 y+4=0\)

Short answer

The equation in standard form is \(\frac{(x-2)^2}{4} + \frac{(y-2)^2}{4} = 1\).

Step-by-step solution

Rearrange the Equation

Start with the given equation: \[x^2 + y^2 - 4x - 4y + 4 = 0\]Move the constant to the other side:\[x^2 - 4x + y^2 - 4y = -4\]

Complete the Square for x-terms

Take the x-terms and complete the square:The expression is \(x^2 - 4x\).Take half of -4 (\(-\frac{4}{2} = -2\)), and square it (\((-2)^2 = 4\)): Add and subtract 4 inside the equation:\[(x^2 - 4x + 4)\]This simplifies to \((x-2)^2\).

Complete the Square for y-terms

Take the y-terms and complete the square:The expression is \(y^2 - 4y\). Take half of -4 (\(-\frac{4}{2} = -2\)), and square it (\((-2)^2 = 4\)): Add and subtract 4 inside the equation:\[(y^2 - 4y + 4)\]This simplifies to \((y-2)^2\).

Write the Completed Squares into the Equation

Insert the completed squares back into the equation:\[(x-2)^2 + (y-2)^2 = -4 + 4 + 4\]Simplify the expression:\[(x-2)^2 + (y-2)^2 = 4\]

Identify the Standard Form of the Ellipse

The equation \((x-2)^2 + (y-2)^2 = 4\) is already in the standard form of an ellipse centered at \((h, k) = (2, 2)\) with \(a^2 = 4\) and \(b^2 = 4\). The equation \(\frac{(x-2)^2}{4} + \frac{(y-2)^2}{4} = 1\) indicates it is a circle, a special case of an ellipse.

Key concepts

Completing the Square

Completing the square is an essential method used to simplify quadratic expressions and put them in a form that is easy to analyze. In the context of converting a general conic equation to its standard form, completing the square is crucial. Here's how it works:

• Identify the quadratic and linear terms for each variable in the equation.
• For instance, when working with the quadratic expression \( x^2 - 4x \), observe the coefficient of \( x \), which is -4.
• Take half of this coefficient: \(-\frac{4}{2} = -2\).
• Square this result to complete the square: \((-2)^2 = 4\).
• Add and subtract this new term (4) inside the expression to rewrite it as a perfect square trinomial. This gives us \((x-2)^2\).

This method is repeated for each variable in the equation, helping to transform it into an easy-to-read format. Completing the square is a powerful algebraic tool used in various areas of mathematics, including solving equations and graphing conics.

Ellipse Equation

An ellipse is a type of conic section that brings together a family of curves defined within a plane. When an ellipse is centered at a point other than the origin, its equation has a unique form. Transforming an ellipse equation to its standard form allows us to understand its properties more clearly.

• The standard form of an ellipse equation with a center at \((h, k)\) is presented as \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).
• Here, \( (h, k) \) is the center of the ellipse.
• The values \( a^2 \) and \( b^2 \) represent squares of semi-major and semi-minor axes respectively. If \( a = b \), then the ellipse becomes a circle.

The process of completing the square helps in rewriting the equation to fit this standard form by isolating and balancing square terms. This neat organization reveals crucial information about the orientation and dimensions of the ellipse.

Geometry of Conics

Conic sections like ellipses represent the intersection of a plane and a double-napped cone. The study of these curves, which also includes hyperbolas, parabolas, and circles, is known as the geometry of conics. Understanding conics is fundamental to both geometry and algebra.

• Each conic section, including ellipses, can be characterized by its unique standard equation and geometric properties.
• The orientation and position are determined by parameters within the equation, such as the center, axes, and relative size.
• Ellipses, in particular, have special attributes, like having two focal points and the sum of the distances from any point on the ellipse to the foci being constant.

Conics have practical applications in various fields such as astronomy, engineering, and physics. For instance, the orbits of planets are elliptical, making the principles of conics vital for understanding celestial motion. Thus, learning to work with and visualize conics is a key step in mastering mathematics.