Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,±4) passes through the point (3,1)

Step-by-Step Solution

Answer

The equation of the ellipse in standard form is \(\frac{x^2}{0.6} + \frac{y^2}{16} = 1\).

Step 1: Identify Values

From the given vertices of the ellipse, 'b' can be determined. Vertices are (0,±4) so 'b' = 4.

Step 2: Substitute the point into the Ellipse Equation

The equation for an ellipse in standard form is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). We know that 'b=4', but 'a' is still unknown. Substitute point (3,1) into the equation and solve for 'a'. This gives \(\frac{3^2}{a^2} + \frac{1^2}{4^2} = 1\).

Step 3: Solve for 'a'

On simplifying the above equation, we get \(\frac{9}{a^2} + \frac{1}{16} = 1\). Solving this equation for 'a^2', gives 'a^2' = 9/15 = 0.6

Step 4: Write the Standard Form of the Ellipse

Now that we have 'a^2' and 'b^2', we can write the standard form of the ellipse, which is \(\frac{x^2}{0.6} + \frac{y^2}{16} = 1\).

Key Concepts

Standard Form

The standard form of the equation for an ellipse gives us a precise mathematical way to describe its shape. It helps in defining the ellipse's attributes such as its axes, vertices, and orientation. The equation of an ellipse in standard form is typically written as:

• For a horizontally oriented ellipse: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
• For a vertically oriented ellipse: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \)

Here, \( (h, k) \) represents the center of the ellipse while \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively. In the context of our exercise, the center is at the origin \( (0,0) \), simplifying the equation to \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, the orientation depends on the lengths of semi-axes \( a \) and \( b \), where \( a > b \) specifies a horizontal orientation, and \( b > a \) gives a vertical orientation.

Vertices

Vertices are key points on an ellipse that define its shape and size. In a typical ellipse, the vertices lie on the major axis, the longest diameter of the ellipse. These points are located at distances of \( \pm a \) and \( \pm b \) from the center, depending on the orientation of the ellipse. For our given exercise, the vertices are \( (0, \pm 4) \), indicating that the major axis is vertical. This tells us that \( b = 4 \). The vertices are crucial for determining the semi-axes since they provide the range of the ellipse along its major axis.

Origin

The origin \( (0,0) \) serves as a common reference point in coordinate geometry. When an ellipse is centered at the origin, it greatly simplifies the equation.The center \( (h,k) \) effectively becomes \( (0,0) \), making the standard equations of the ellipse easier to manage:

• Without shifts: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

In the exercise at hand, the ellipse is centered at \( (0,0) \), reducing its equation to the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Figuring out the specific values for \( a \) and \( b \) allows us to completely describe the ellipse's geometry.

Conic Sections

Ellipses are one type of conic section, a curve formed by the intersection of a cone with a plane. These curves include circles, ellipses, parabolas, and hyperbolas. An ellipse is created when the intersecting plane cuts through a right circular cone at an angle, resulting in a closed curve. Here are a few key points about ellipses as conic sections:

• An ellipse might appear circular but is distinguished by a stretched, oval shape.
• The defining attribute is the consistent sum of distances from any point on the ellipse to two fixed points called foci.
• Ellipses have practical applications in astronomy, orbital mechanics, and various fields of engineering.

Understanding the role and characteristics of ellipses within conic sections helps in comprehending their mathematical significance and real-world applications.