Problem 43 - College Algebra

Question

For the following exercises, find the foci for the given ellipses. $$ 16 x^{2}+64 x+4 y^{2}-8 y+4=0 $$

Step-by-Step Solution

Answer

The foci are at \((-2, 1 \pm 2\sqrt{3})\).

Step 1: Rearrange the Equation

Starting with the given equation: \[ 16x^2 + 64x + 4y^2 - 8y + 4 = 0 \] we need to rearrange it into the standard form of an ellipse. First, move the constant to the other side: \[ 16x^2 + 64x + 4y^2 - 8y = -4 \]

Step 2: Complete the Square for x-terms

Focus on the x-terms first: \[ 16x^2 + 64x \] Factor out a 16: \[ 16(x^2 + 4x) \] To complete the square, take half of 4 (the coefficient of x), square it to get 4, and adjust inside the bracket: \[ 16(x^2 + 4x + 4 - 4) \] which simplifies to: \[ 16((x + 2)^2 - 4) = 16(x + 2)^2 - 64 \]

Step 3: Complete the Square for y-terms

Now focus on the y-terms: \[ 4y^2 - 8y \] Factor out a 4: \[ 4(y^2 - 2y) \] To complete the square, take half of -2 (the coefficient of y), square it to get 1, and adjust inside the bracket: \[ 4(y^2 - 2y + 1 - 1) \] which simplifies to: \[ 4((y - 1)^2 - 1) = 4(y - 1)^2 - 4 \]

Step 4: Combine and Simplify the Equation

Combine the completed squares from Steps 2 and 3: \[ 16(x + 2)^2 - 64 + 4(y - 1)^2 - 4 = -4 \] Simplify: \[ 16(x + 2)^2 + 4(y - 1)^2 = 64 \] Divide every term by 64 to get the standard ellipse form: \[ \frac{(x + 2)^2}{4} + \frac{(y - 1)^2}{16} = 1 \] This is now in the standard form \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \).

Step 5: Identify the Parameters

From the standard form \( \frac{(x+2)^2}{4} + \frac{(y-1)^2}{16} = 1 \) of the ellipse, identify: - The center at \((h, k) = (-2, 1)\). - \(a^2 = 4\) (semi-major axis along the y-axis) and \(b^2 = 16\) (semi-minor axis along the y-axis). Since \(b^2 > a^2\), the major axis is vertical.

Step 6: Calculate the Foci of the Ellipse

To find the foci, use the formula \(c^2 = b^2 - a^2\): \[ c^2 = 16 - 4 \] \[ c^2 = 12 \] \[ c = \sqrt{12} = 2\sqrt{3} \] Since the major axis is vertical, the foci are located at \((h, k \pm c)\), which gives: \((-2, 1 \pm 2\sqrt{3})\).

Key Concepts

Completing the Square

Completing the square is a critical algebraic technique used to manipulate quadratic equations into a more useful form. This method is especially applicable when dealing with conic sections like ellipses. When approaching the process, focus on finding expressions in the form

• \( (x + p)^2 \)
• \( (y + q)^2 \)

To successfully complete the square:

• Look at the quadratic term and the linear term together, as seen in the expressions with \( x^2 \) and \( x \), or \( y^2 \) and \( y \).
• Factor out any leading coefficient from the quadratic term, simplifying the expression inside the parentheses.
• Take half of the coefficient of the linear term, square it, and modify the expression. Adjust by adding and subtracting this value within the parentheses to create a perfect square trinomial.

Completing the square allows the quadratic expression to be rewritten in a form that reveals the essential characteristics of the ellipse.

Foci of an Ellipse

The foci of an ellipse are two special points along its major axis. They are instrumental in the geometric definition of the ellipse. An ellipse can be defined as the collection of all points where the sum of the distances to the two foci is constant. To determine the foci:

• Identify the lengths of the major and minor axes from the equation in standard form, represented by \( a^2 \) and \( b^2 \).
• The foci are located along the major axis, and their distance "\( c \)" from the center is given by the equation \( c^2 = b^2 - a^2 \) when \( b > a \), or \( c^2 = a^2 - b^2 \) when \( a > b \).
• Calculate "\( c \)" and find the foci at positions \((h \pm c, k)\) or \((h, k \pm c)\) depending on whether the major axis is horizontal or vertical.

The foci help to understand the shape and orientation of the ellipse, providing insight into its geometric properties.

Standard Form of an Ellipse

The standard form of an ellipse is a concise way of expressing ellipses that reveals their key attributes, such as the center, axes, and geometry. The equation format is given by \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]where

• \( (h, k) \) is the center of the ellipse.
• \( a^2 \) and \( b^2 \) are the squares of the semi-major and semi-minor axes, respectively.
• The ellipse is aligned with the coordinate axes, meaning the axes of the ellipse are either parallel to the x or y-axis.

If \( a^2 > b^2 \), the ellipse is horizontally stretched, making the major axis parallel to the x-axis. Conversely, if \( b^2 > a^2 \), the ellipse is vertically stretched. The standard form is critical for calculating properties like the axes length and finding the foci.

Conic Sections

Conic sections represent the curves obtained by intersecting a plane with a cone. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each type is characterized by specific geometric and algebraic properties. An ellipse is one such figure and is defined when the plane cuts through both nappes of the cone at an angle, producing a closed, rounded shape. The distinguishing features include:

• A major and minor axis, indicating the ellipse's width and height.
• A center, where the axes intersect.
• Two foci, which provide a constant sum of distances to any point on the ellipse.

Conic sections are fundamental in mathematics because they model various physical phenomena and are used extensively in physics, engineering, and astronomy. Understanding the principles of conic sections creates a foundation for exploring more complex geometrical constructs and their applications.