Problem 28 - College Algebra

Question

For the following exercises, find the foci for the given ellipses. $$ \frac{(x+1)^{2}}{100}+\frac{(y-2)^{2}}{4}=1 $$

Step-by-Step Solution

Answer

Foci: \((-1 - 4\sqrt{6}, 2)\) and \((-1 + 4\sqrt{6}, 2)\).

Step 1: Identify the Form of the Equation

The given ellipse equation is \( \frac{(x+1)^{2}}{100}+\frac{(y-2)^{2}}{4}=1 \). It is of the form \( \frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \), where \( (h, k) \) is the center. Here, the center is \((-1, 2)\).

Step 2: Determine Semi-major and Semi-minor Axes

The equation \( \frac{(x+1)^{2}}{100}+\frac{(y-2)^{2}}{4}=1 \) has denominators 100 and 4. Since \( 100 > 4 \), the semi-major axis lies along the x-direction with \( a^2=100 \) (hence \( a=10 \)) and \( b^2 = 4 \) (hence \( b=2 \)).

Step 3: Calculate the Distance to Foci

For an ellipse, the distance \( c \) from the center to each focus is given by \( c = \sqrt{a^2 - b^2} \). Calculate \( c = \sqrt{100 - 4} = \sqrt{96} = 4\sqrt{6} \).

Step 4: Determine the Coordinates of the Foci

Since the ellipse's major axis is along the x-axis, shift the center left and right by \( c \) along the x-axis to find the foci. The coordinates of the foci are \( (-1 - 4\sqrt{6}, 2) \) and \( (-1 + 4\sqrt{6}, 2) \).

Key Concepts

Ellipse Equation

An ellipse is a geometric shape that looks like an elongated circle, and its equation has a specific standard form. For this specific exercise, we deal with the equation: \[ \frac{(x+1)^2}{100} + \frac{(y-2)^2}{4} = 1 \]The general form of an ellipse equation is:\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]Here,

• \((h, k)\) is the center of the ellipse.
• \(a^2\) and \(b^2\) represent squared values of the semi-major and semi-minor axes, respectively.

In our equation, the center is at point \,(-1, 2), as derived by comparing it to the general form, where \((x + 1)\) and \((y - 2)\) indicate shifts to the center.

Foci of Ellipse

The foci of an ellipse are two special points inside the ellipse, positioned along the major axis. They have a fundamental property that any point on the ellipse has the same total distance to these two foci.To find the foci, we calculate the distance from the center to the foci, denoted as \(c\), using the formula:\[ c = \sqrt{a^2 - b^2} \]In this case:

• \(a^2 = 100\) and \(b^2 = 4\)
• So, \(c = \sqrt{100 - 4} = \sqrt{96} = 4\sqrt{6}\).

The foci are then determined by adjusting the center along the major axis, which is aligned horizontally for this equation.

Semi-Major and Semi-Minor Axes

These axes are crucial in defining the ellipse's shape.

• The semi-major axis is the longest radius of the ellipse, whereas the semi-minor axis is the shortest.
• For the given equation, since \(100 > 4\), it tells us that the semi-major axis is along the x-axis, and its value is \(a = 10\).
• The semi-minor axis lies along the y-direction with \(b = 2\).

This aligns with our general understanding, where the larger denominator indicates the direction of the semi-major axis.

Distance to Foci

Understanding the distance to the foci helps in plotting and describing the ellipse more accurately. The calculation for \(c\), the distance from the center of the ellipse to each focus, uses the difference between the squares of the semi-major and semi-minor axes:\[ c = \sqrt{a^2 - b^2} \]After computation, we find:

• \(c = \sqrt{96} = 4\sqrt{6}\)
• This distance describes how far each focus is from the center \((-1, 2)\).

By adding and subtracting \(c\) from the x-coordinate of the center, we get the actual foci coordinates as \((-1 - 4\sqrt{6}, 2)\) and \((-1 + 4\sqrt{6}, 2)\).