Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the eccentricity of the ellipse. $$x^{2}+9 y^{2}-10 x+36 y+52=0$$

Step-by-Step Solution

Answer

The eccentricity of the given ellipse is \(\frac{3}{5}\).

Step 1: Convert the Equation into Standard Form

The standard form of an ellipse equation is written in the form \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\), where \((h,k)\) is the center of the ellipse and \(a\) and \(b\) are the lengths of the semi-axes. In this case, we need to group all \(x\) terms and \(y\) terms to complete the square and get the standard form:\[x^{2}-10x+9 y^{2}+36 y+52=0 \rightarrow (x^{2}-10x) + (9y^{2}+36 y) = -52 \rightarrow\(x-5)^{2} -25+ (9(y+2)^{2}-36) = -52\rightarrow \frac{(x-5)^{2}}{25}+\frac{(y+2)^{2}}{4}=1\] So, \(a^{2}=25\) and \(b^{2}=4\).

Step 2: Calculate Eccentricity

The eccentricity, \(e\), is calculated by the formula \(e=\sqrt{1-\frac{b^{2}}{a^{2}}}=\sqrt{1-\frac{4}{25}}=\frac{3}{5}\). So, the eccentricity of the given ellipse is \(\frac{3}{5}\).

Key Concepts

Ellipse Equation

An ellipse is a geometric shape that can be described using an equation. The general form of an ellipse equation involves terms containing both of the variables:

• For an ellipse centered at the origin with horizontal and vertical axes, the equation is: \( x^2 + y^2 = c \).
• When translated, the ellipse equation becomes: \( x^2 + (k)y^2 = l \), where the coefficients of the squared terms differ.
• For generic non-standard forms, additional terms involving \(x\) and \(y\) may exist and need to be rearranged.

In this particular problem, we started with an equation not in the standard ellipse form. Our goal is to rearrange it into a more recognizable format by completing operations like grouping and completing the square.

Completing the Square

Completing the square is pivotal when transforming the general equation of an ellipse into its canonical or standard form. Here's a short guide:

• Identify and group \(x\)-related and \(y\)-related terms: It allows for separate treatment of each variable.
• Complete the square for each group:
  • For any binomial \(x^2 - bx\), add and subtract \(\left(\frac{b}{2}\right)^2\) to complete the square.
  • Similarly, for \(y^2 + dy\), add and subtract \(\left(\frac{d}{2}\right)^2\).

In this exercise, for example, the transformation of \(x^2 - 10x\) becomes \((x-5)^2 - 25\). This step is crucial to rewriting the initial equation into an understandable form where ellipse properties can be easily identified.

Calculation of Eccentricity

The eccentricity of an ellipse is a measure of how much it deviates from being a perfect circle. For ellipses, it's always between 0 and 1:

• Formula for eccentricity \(e\) is: \[ e = \sqrt{1 - \frac{b^2}{a^2}} \]where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.
• The semi-major axis is the longest radius of the ellipse.
• In our example, since \(a^2=25\) and \(b^2=4\), calculating \(e\) yields \(\sqrt{1 - \frac{4}{25}} = \frac{3}{5}\).

This step helps quantify the shape's elongation, quite relevant in applications like orbital dynamics and optics.

Standard Form of Ellipse

The standard form of an ellipse equation is invaluable for recognizing its key features, the center and axes lengths:

• Written generally as: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \], indicating a centered ellipse at \((h,k)\).
• The terms \((x-h)\) and \((y-k)\) reveal shifts from the origin, efficiently pinpointing the ellipse's position.
• \(a^2\) under the squared \(x\)-term defines the direction and length of the major axis, while \(b^2\) under \(y\)-term determines the minor axis.

Achieving this form from completion of the square allows straightforward identification of ellipse dimensions and eccentricity, enriching conceptual comprehension of the shape's geometry.