Problem 56 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given characteristics. focus \((0,3 \sqrt{2}),\) height 19

Step-by-Step Solution

Answer

The equation of the ellipse in standard form with center at the origin, a focus at (0, 3√2), and a height of 19, is \(x^2/((9.5)^2 - (3√2)^2) + y^2/(9.5)^2 = 1\).

Step 1: Calculation of 'a' and 'b'

With the provided characteristics of the ellipse, the distance from the center to a vertex, a, is half the height of the ellipse. So, \(a = 19/2 = 9.5\). The distance from the center to the focus, b, is the y-coordinate of the given focus. So, \(b = 3√2\).

Step 2: Calculation of 'c'

Next, you need to find 'c', the distance between the center and the directrix. For an ellipse, the relation among a, b and c is \(a^2 = b^2 + c^2\) Therefore, \(c^2 = a^2 - b^2 = (9.5)^2 - (3√2)^2\).

Step 3: Forming the equation

With a, b, and c calculated, the equation of the ellipse becomes \((x-0)^2/c^2 + (y-0)^2/a^2 = 1\) which simplifies to \(x^2/c^2 + y^2/a^2 = 1\). Substitute the calculated values of a and c to find the equation of the ellipse.

Key Concepts

Ellipse Properties

Ellipses are fascinating geometric shapes that resemble stretched circles. To fully grasp ellipses, you need to understand some key properties. An ellipse has two focal points, and the sum of the distances from any point on the ellipse to these foci is constant.
This property is what gives an ellipse its unique shape, different from circles. In addition to the foci, there are vertices—these are the points on the longest diameter of the ellipse.
Understanding the terms is crucial:\( \: \)

• Major Axis: The longest diameter of the ellipse, passing through the foci and vertices.
• Minor Axis: The shortest diameter, perpendicular to the major axis at the center.
• Center: The midpoint of both the major and minor axes.

Not only do ellipses look elegant, but they also have interesting optical properties. For instance, a whisper in one focus can be heard clearly at the other, because of how the sound waves reflect.

Standard Form of an Ellipse

The standard form of an ellipse is a way to write its equation that clearly highlights its geometric properties. When you hear "standard form," you should think about simplicity and order, making it easier to interpret the ellipse's main features.
The standard form equation of an ellipse with center \((h, k)\) is: \[ \frac{{(x-h)^2}}{a^2} + \frac{{(y-k)^2}}{b^2} = 1 \] Here’s how you can understand the components:

• \(a\): The semi-major axis, which is half the length of the major axis.
• \(b\): The semi-minor axis, which is half the length of the minor axis.

For an ellipse in standard form, if \(a > b\), the ellipse is wider than it is tall; if \(b > a\), it is taller than it is wide. Notice how the formula sets \(a\) always with the larger axis, ensuring consistency when reading or writing an ellipse's equation.

Ellipse with Center at Origin

When an ellipse's center is located at the origin \((0, 0)\), the equation simplifies significantly. It’s a special and commonly encountered case in mathematics.
The standard form of the equation for such an ellipse is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] This form is very convenient because the terms \((x-0)^2\) and \((y-0)^2\) reduce to \(x^2\) and \(y^2\), respectively. When we take the specific example of this ellipse with center at the origin, we can see:

• Axis Alignment: The axes of the ellipse align perfectly with the Cartesian coordinate axes, meaning it reaches its maximum and minimum extents directly along the axes.
• Simplicity: Easy to graph and interpret, as there is no horizontal or vertical translation required.

Knowing how to handle ellipses centered at the origin helps immensely in both solving problems and utilizing ellipses in real-world applications, like optic design and astronomy.