Problem 18 - Algebra and Trigonometry

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ 6 x^{2}=30-5 y^{2} $$

Step-by-Step Solution

Answer

The equation represents an ellipse centered at the origin with a semi-major axis of length \(\sqrt{6}\), semi-minor axis of length \(\sqrt{5}\), and the foci located at points \((-1,0)\) and \((1,0)\).

Step 1: Conversion to Standard Form

Let's first put the given equation into the standard form. Divide all terms by 30 to normalize the equation: \(\frac{6x^2}{30} + \frac{5y^2}{30} = \frac{30}{30}\). Simplifying, we obtain \(0.2x^{2} - 0.167y^{2} = 1\).

Step 2: Identifying the Ellipse Parameters

From the standard form, we can identify the parameters \(a\) and \(b\), which correspond to the values of \(x\) and \(y\) when the equation equals one. We have \(a^2 = 0.2^{-1} = 5\) and \(b^2 = 0.167^{-1}\approx 6\).

Step 3: Calculate the Foci

The foci of an ellipse are at a distance of \(c = \sqrt{|a^2 - b^2|}\) from the center. Using the values of \(a^2\) and \(b^2\), calculated in the previous step, we find \(c = \sqrt{|5 - 6|} = \sqrt{1}= 1\). The foci are \(1\) in either direction from the center of the ellipse, which is at the origin.

Step 4: Plot the Ellipse

The plot is a standard ellipse centered at origin with semi-major axis of length \(\sqrt{6}\), semi-minor axis of length \(\sqrt{5}\), and the foci at \((-1,0)\) and \((1,0)\).

Key Concepts

Foci of an Ellipse

The foci of an ellipse are specific points located along its major axis. They hold a special property: any point on the ellipse maintains the sum of distances to these two foci constant. Understanding this helps in sketching the ellipse accurately and solving related problems.

To find the foci when the ellipse is centered at the origin, we can use the formula for the distance from the center to each focus: \[c = \sqrt{|a^2 - b^2|}\]*where \(a^2\) and \(b^2\) are the squares of the semi-major and semi-minor axes, respectively.*

For the given exercise, once we determined that \(a^2 = 5\) and \(b^2 = 6\), applying the formula gives \(c = 1\). This means the foci are precisely one unit from the origin along the x-axis. Thus, the coordinates of the foci are \((-1, 0)\) and \((1, 0)\). Knowing the foci's position is essential for graphing the ellipse and understanding its structure.

Standard Form of Ellipse

The standard form of an ellipse’s equation is crucial for identifying its properties, like the center, axes, and orientation. It appears as follows for an ellipse centered at the origin:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]If the ellipse is not centered at the origin, we'll include coordinates \(h\) and \(k\) for the center, in this manner: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]

In the exercise, the given equation was not initially in standard form. We converted it by dividing through by the constant, resulting in: \[0.2x^2 - 0.167y^2 = 1\]This unveils the values of \(a^2\) and \(b^2\). The formula rearranges the equation to help easily spot the lengths of axes and locate the center. Understanding this setup allows students to readily graph the ellipse and extract information about its structure.

Ellipse Parameters

The ellipse parameters include critical measurements like its axes, center, and eccentricity. These measurements help us determine the shape and size of the ellipse. Every ellipse has two main axes:

• Semi-major axis: The longest radius of the ellipse, stretching from the center to the perimeter, represented by \(a\).
• Semi-minor axis: The shortest radius, perpendicular to the semi-major, represented by \(b\).

Once the equation of an ellipse is in its standard form, multiplying \(a\) or \(b\) by 2 gives the full lengths of the axes.

From our exercise, the semi-major and semi-minor axes lengths are \(\sqrt{6}\) and \(\sqrt{5}\) respectively. These axes define the ellipse's stretch in the x and y directions. Additionally, for an ellipse centered on the origin without horizontal/vertical shift, determining \(c\) allows us to map its foci, further defining its spatial boundaries.Understanding these parameters is paramount when graphing or analyzing any ellipse, offering clarity on its geometrical properties.