Problem 31 - Algebra and Trigonometry

Question

Find an equation for the ellipse that satisfies the given conditions. Length of major axis \(4,\) length of minor axis \(2,\) foci on \(y\) -axis

Step-by-Step Solution

Answer

The equation of the ellipse is: \( x^2 + \frac{y^2}{4} = 1 \).

Step 1: Understand the Orientation

Because the foci are on the \(y\)-axis, the ellipse is vertically oriented. This means the major axis is vertical, and the general equation of the ellipse is: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \] where \(a > b\).

Step 2: Determine Values of \(a\) and \(b\)

The length of the major axis is \(4\), which means \(2a = 4\). Thus, \(a = 2\). The length of the minor axis is \(2\), which implies \(2b = 2\). Therefore, \(b = 1\).

Step 3: Write the Equation of the Ellipse

Substitute \(a\) and \(b\) into the ellipse equation: \[ \frac{x^2}{1^2} + \frac{y^2}{2^2} = 1 \] Simplify it to get: \( x^2 + \frac{y^2}{4} = 1 \).

Step 4: Verify the Conditions

Check that the obtained equation satisfies all given conditions. The equation \( x^2 + \frac{y^2}{4} = 1 \) matches the conditions of having the major axis along the \(y\)-axis with the correct lengths for the axes. This verifies the solution as correct.

Key Concepts

Major Axis

The major axis of an ellipse is the longest diameter, which passes through the center and both foci of the ellipse. It is an essential component in describing the size and shape of the ellipse.
For this particular exercise, the length of the major axis is given as 4. Since the length of an axis is defined as twice its semi-length, the semi-major axis is half of the major axis length: \(a = \frac{4}{2} = 2\).

• Length of the major axis = 4
• Semi-major axis, \(a = 2\)

Having a clear understanding of the major axis helps in determining the orientation of the ellipse, as it indicates which axis is longer and consequently more dominant in the shape's configuration.

Minor Axis

The minor axis is the shorter diameter of the ellipse, perpendicular to the major axis and also passing through the center. In this exercise, the length of the minor axis is identified as 2, which means the semi-minor axis (half the minor axis) is 1.

• Length of the minor axis = 2
• Semi-minor axis, \(b = 1\)

Understanding the minor axis is crucial for writing the equation of the ellipse and analyzing its overall dimensions. It helps in balancing the elongated shape signified by the major axis.

Foci

Foci (plural of focus) are two specific and vital points located along the major axis within an ellipse. They are crucial because the total distance from any point on the ellipse to the two foci is constant, defining the ellipse's unique shape.
In this case, since the foci are on the \(y\)-axis, it confirms that the major axis is also vertical. The distances between the center and each focus can be determined from the relationship \(c^2 = a^2 - b^2\):\[c^2 = 2^2 - 1^2 = 4 - 1 = 3\]Therefore, the distance \(c = \sqrt{3}\). The foci are thus located at \((0, \sqrt{3})\) and \((0, -\sqrt{3})\) along the \(y\)-axis.

• Foci are points on the major axis at \((0, \pm \sqrt{3})\)
• Total distance from any ellipse point to foci is constant

Recognizing the position of foci helps in solidifying the orientation and equation of the ellipse.

Orientation of Ellipse

The orientation of an ellipse defines how it is positioned relative to the coordinate axes. It is determined by whether the major axis aligns with the x-axis or y-axis.
In the given exercise, the foci are located on the \(y\)-axis, indicating a vertical orientation. This means the major axis is vertical:

• Vertical orientation
• Major axis along the \(y\)-axis, indicating \(a\) is associated with \(y\)

The equation for a vertical ellipse is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), with \(a > b\). With \(a = 2\) and \(b = 1\), the final equation is \(x^2 + \frac{y^2}{4} = 1\). Understanding the alignment and using it to express the equation correctly is pivotal for solving problems involving ellipses.