Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the eccentricity of the ellipse. $$4 x^{2}+3 y^{2}-8 x+18 y+19=0$$

Step-by-Step Solution

Answer

The eccentricity of the ellipse is \(\sqrt{2/3}\)

Step 1: Rewrite the equation in standard form

Divide the given equation by 4 and complete the squares to move it into a form that resembles the standard equation for an ellipse. The equation becomes: \(4(x - 1)^2/4 + 9(y + 1)^2/12 = 1\)

Step 2: Identify the center and semi-axes

From the standard form, we can find the center of the ellipse is at (1, -1), and \(a^2=4, b^2=12\). So, \(a=2\) is the semi minor axis and \(b=\sqrt{12}\) is the semi major axis.

Step 3: Compute the eccentricity

Substitute \(a\) and \(b\) into the formula \(\sqrt{1 - (a^2 / b^2)}\) which yields \(e = \sqrt{1 - (4/12)} = \sqrt{2/3}\).

Key Concepts

Standard Form of an Ellipse

To solve problems involving ellipses, understanding their equation's standard form is essential. The standard form of an ellipse's equation provides a clear framework, written as \[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]where

• \((h, k)\) represents the center of the ellipse.
• \(a\) and \(b\) are the lengths of the semi-axes.

Based on this structure, when dealing with an equation like \[4x^2 + 3y^2 - 8x + 18y + 19 = 0,\]we need to manipulate it into this standard form by performing operations such as completing the square and making sure the right-hand side equals 1. This form makes it easy to identify the crucial elements of the ellipse, such as its center and the lengths of its semi-major and semi-minor axes.

Completing the Square

Completing the square is a vital algebraic technique used to transform a quadratic expression into a perfect square trinomial. This process is crucial for rewriting the equation of an ellipse in standard form.Let's look at how completing the square helps with an equation like \[4x^2 + 3y^2 - 8x + 18y + 19 = 0.\]

• First, separate the equation into parts containing \(x\) and \(y\): \(4(x^2 - 2x) + 3(y^2 + 6y) = -19\).
• To complete the square for \(x\), take half of \(-2\) (which is \(-1\)), square it, and adjust the equation: \((x - 1)^2\). Similarly, for \(y\), take half of \(6\) (which is \(3\)), square it, and adjust the term: \((y + 3)^2\).

This gives us \[4(x - 1)^2 + 3(y + 1)^2 = -19 + 4 + 27\].Then, simplify and finally divide by the constant to set the equation to 1, finishing the square and aiding in transitioning to the standard ellipse form.

Ellipse Center

The center of an ellipse is a critical concept in defining its position and ensuring a proper understanding of its geometry. In the ellipse's standard form equation:\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1,\]\((h, k)\) represents the ellipse center. Identifying this point provides insights into the symmetry and orientation of the ellipse relative to the Cartesian coordinate system.For example, if you successfully convert an equation like \[4(x - 1)^2/4 + 9(y + 1)^2/12 = 1\]into standard form, the terms

• \((x - 1)\) give an \(h\) value of 1,
• and \((y + 1)\) give a \(k\) value of -1.

Thus, placing the center of the ellipse at \((1, -1)\). Knowing the center aids in graphing and understanding the ellipse's relationship with other geometric figures.

Semi-major and Semi-minor Axes

The semi-major and semi-minor axes are the backbone of an ellipse's structure, defining its size and shape. In the equation's standard form:\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1,\]the terms \(a^2\) and \(b^2\) are crucial.

• When \(a^2\) is less than \(b^2\), \(a\) is the semi-minor axis and represents the shortest radius of the ellipse.
• Conversely, \(b\) becomes the semi-major axis, representing the longest radius.

Example: After converting the initial equation \[4(x - 1)^2/4 + 9(y + 1)^2/12 = 1\]we find \(a^2 = 4\) and \(b^2 = 12\).

• Solving gives \(a = 2\) and the semi-minor axis.
• \(b = \sqrt{12}\) indicates the semi-major axis.

Grasping these axes' roles enhances understanding of the ellipse's geometry and allows for correct depiction of its size and proportions in visual or analytical tasks.