Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$4 x^{2}-16 x+9 y^{2}+36 y=-16$$

Step-by-Step Solution

Answer

The graph of the equation is an ellipse with a horizontal axis.

Step 1: Rewrite the Equation

Begin with the given equation: \[4x^2 - 16x + 9y^2 + 36y = -16\]Rearrange it to move constants to the other side, which results in:\[4x^2 - 16x + 9y^2 + 36y + 16 = 0\]

Step 2: Completing the Square for x-terms

Focus on the x-terms in the equation: \[4x^2 - 16x\]Factor out the 4:\[4(x^2 - 4x)\]To complete the square, take half of the coefficient of x, square it, and add inside the parenthesis:\[4(x^2 - 4x + 4 - 4)\]This can be rewritten as:\[4((x-2)^2 - 4)\]

Step 3: Completing the Square for y-terms

Now focus on the y-terms: \[9y^2 + 36y\]Factor out the 9:\[9(y^2 + 4y)\]To complete the square, take half of the coefficient of y, square it, and add inside the parenthesis:\[9(y^2 + 4y + 4 - 4)\]This becomes:\[9((y+2)^2 - 4)\]

Step 4: Substitute Complete Squares into the Equation

Replace the complete squares from Steps 2 and 3 back into the equation:\[4((x-2)^2 - 4) + 9((y+2)^2 - 4) + 16 = 0\]Simplify the equation:\[4(x-2)^2 - 16 + 9(y+2)^2 - 36 + 16 = 0\]Which simplifies further to:\[4(x-2)^2 + 9(y+2)^2 = 36\]

Step 5: Divide by Constant to Identify the Conic Section

Divide every term by 36 to simplify:\[\frac{4(x-2)^2}{36} + \frac{9(y+2)^2}{36} = 1\]Simplify the fractions:\[\frac{(x-2)^2}{9} + \frac{(y+2)^2}{4} = 1\]

Step 6: Identify the Conic Section

The simplified equation \(\frac{(x-2)^2}{9} + \frac{(y+2)^2}{4} = 1\) is an equation of an ellipse. This is because the sum of two squared terms equals 1, with different denominators indicating an ellipse. The larger denominator (9) under \((x-2)^2\) suggests a horizontal major axis.

Key Concepts

Ellipse Equation

The ellipse equation is a fascinating member of the conic sections family. An ellipse is essentially a stretched circle, with an elongated shape depending on its axes lengths. The standard form of an ellipse equation is: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). This equation represents an ellipse centered at the point \((h, k)\).

• The terms \((x-h)^2\) and \((y-k)^2\) are squared terms, indicating the form of an ellipse.
• The denominators \(a^2\) and \(b^2\) represent the squares of the distances of the semi-major and semi-minor axes.
• If \(a > b\), the semi-major axis is horizontal; if \(b > a\), it is vertical.

This equation highlights that the sum of squared terms equals 1, which is distinct for ellipses in conic sections.

Completing the Square

Completing the square is a crucial algebraic method used to transform a quadratic expression into a perfect square trinomial. This technique enables the reformation of expressions, simplifying them down to their essential components for easier interpretation or further calculations. To complete the square:

• Begin with a quadratic expression in the form \(ax^2 + bx\).
• Using an intermediate term, \(c\), form a perfect square trinomial \((x+d)^2\).
• Calculate \(d\) by taking half of the \(b\) term, then squaring the result.

When applied to conic sections, completing the square is integral in reformulating equations into a more recognizable form, resembling the standard ellipse formula.

Horizontal Major Axis

Within the study of ellipses, determining the orientation of the major axis is crucial. The axis determines the direction in which the ellipse is elongated. In our example, the simplified equation is: \[ \frac{(x-2)^2}{9} + \frac{(y+2)^2}{4} = 1 \]

• Here, the term under \((x-2)^2\) is greater than the term under \((y+2)^2\).
• This indicates that the major axis is horizontal because the larger axis corresponds to the larger denominator.
• Geometrically, this means the ellipse stretches out more along the x-axis than the y-axis.

The horizontal major axis emphasizes the broader spread along the horizontal plane, as opposed to a vertical major axis which would result in a taller, narrower shape.