Question

Write the equation of the given ellipse in standard form. \(3 x^{2}+2 y^{2}-12 y+6=0\)

Short answer

The standard form is: \( \frac{x^2}{4} + \frac{(y - 3)^2}{6} = 1 \).

Step-by-step solution

Move the constant to the other side

Start by moving the constant term in the equation to the right side. Subtract 6 from both sides of the equation to get: \[ 3x^2 + 2y^2 - 12y = -6 \]

Arrange terms

To make it easier to complete the square, arrange the terms so like terms are grouped together. \[ 3x^2 + (2y^2 - 12y) = -6 \]

Complete the square for the y-terms

Factor the coefficient of \( y^2 \) out of the y-terms inside the parentheses: \[ 3x^2 + 2(y^2 - 6y) = -6 \]Now, complete the square inside the parentheses:The term to complete the square is \( (\frac{6}{2})^2 = 9 \). Add and subtract this value inside the parentheses:\[ 3x^2 + 2(y^2 - 6y + 9 - 9) = -6 \]Simplify to:\[ 3x^2 + 2((y - 3)^2 - 9) = -6 \]

Simplify and rearrange the equation

Distribute the 2 in the y-terms:\[ 3x^2 + 2(y - 3)^2 - 18 = -6 \]Add 18 to both sides to simplify:\[ 3x^2 + 2(y - 3)^2 = 12 \]

Divide all terms by 12

To express the equation in standard form, divide every term by 12:\[ \frac{3x^2}{12} + \frac{2(y - 3)^2}{12} = \frac{12}{12} \]Simplify each term:\[ \frac{x^2}{4} + \frac{(y - 3)^2}{6} = 1 \]

Write the equation in standard form

The standard form of an ellipse equation is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \( (h, k) \) is the center and \( a^2, b^2 \) are the denominators of the fractions under the squared terms. Thus, the standard form is:\[ \frac{x^2}{4} + \frac{(y - 3)^2}{6} = 1 \]

Key concepts

Standard Form of Ellipse

To express the equation of an ellipse in its standard form, it is essential to understand the general structure of an ellipse equation. The standard form for an ellipse centered at \(h, k\) is given by:
\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]This format helps identify key features of the ellipse:

• \(h\) and \(k\) represent the horizontal and vertical shifts of the center from the origin.
• \(a^2\) and \(b^2\) are the squares of the semi-major and semi-minor axes, representing the half-lengths of the ellipse's axes.

For horizontal ellipses, \(a^2 > b^2\); for vertical ones, \(b^2 > a^2\). This distinction is crucial in determining the orientation of the ellipse.
Completing the square is often a necessary step to manipulate the given equation into this form, enabling the identification of the ellipse's axes and center.

Completing the Square

Completing the square is an essential algebraic technique used to transform quadratic expressions into perfect square trinomials. This is particularly useful when rewriting conic sections, such as ellipses, into their standard forms.
In our exercise, we started with the equation:
\[ 3x^2 + 2y^2 - 12y + 6 = 0 \]During the solution process, we focused on the \(y\) terms to complete the square. Here’s how we proceed with \(y^2 - 6y\):

• First, factor out the coefficient of \(y^2\), which is \2\, to simplify the term inside the parentheses: \2(y^2 - 6y)\.
• Next, calculate \( \left(\frac{6}{2}\right)^2 = 9\) and add and subtract it inside the parentheses to form a perfect square trinomial: \(y^2 - 6y + 9 - 9\).
• This helps to reformulate it as a squared term: \( (y-3)^2\).

Through these steps, the \(y\) part of the ellipse equation transitions into a form suitable for completing the ellipse’s standard form.

Coordinate Geometry

Coordinate geometry revolves around the representation and manipulation of geometric shapes in the Cartesian coordinate system. When dealing with ellipses, coordinate geometry provides tools to interpret and draw these figures based on algebraic equations.
For our problem, identifying the ellipse's core components—center, axes lengths, and orientation—becomes straightforward once the equation is in standard form:

• The origin's offsets are determined by \(h, k\) values in the equation, signifying where the ellipse's center lies in the coordinate plane.
• The values under the squared terms, \(a^2\) and \(b^2\), indicate the extent of the ellipse along its respective axes.
• Understanding these parameters allows us to accurately plot the ellipse, showcasing its horizontal or vertical orientation.

Thus, mastery of coordinate geometry enables a clearer depiction and understanding of ellipses and similar geometric entities.

Algebraic Manipulation

Algebraic manipulation involves the strategic rearrangement and simplification of equations to achieve a desired form or result. In the context of this exercise, algebraic manipulation was crucial in transforming the given quadratic equation into the standard form of an ellipse.
Steps to transform such equations include:

• Reorganizing terms to facilitate completing the square, such as separating like terms.
• Distributing and factoring coefficients to isolate and simplify expressions.
• Dividing the entire equation by a constant to match the ellipse's standard form requirement that the right-hand side equals 1.

The goal is to convert complex, cluttered equations into more intuitive forms that reveal the geometric characteristics of the shapes they describe. In doing so, one unlocks a deeper understanding of the relationship between algebraic expressions and their geometric interpretations.