Problem 5 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given vertex and co-vertex. $$ (0,-7),(4,0) $$

Step-by-Step Solution

Answer

The equation of the ellipse in standard form with center at the origin and given vertex and co-vertices is \[\frac{x^2}{16}+\frac{y^2}{49}=1\]

Step 1: Identify 'a' and 'b'

Given the vertex (0,-7) and co-vertex (4,0), it is understood that 'b' = 7 (from the vertex) and 'a' = 4 (from the co-vertex).

Step 2: Substitute 'a' and 'b' values in equation

Substitute 'a' = 4 and 'b' = 7 into the standard form equation of an ellipse. The equation comes out as \[ \frac{x^2}{4^2}+\frac{y^2}{7^2}=1 \].

Step 3: Simplify the equation

Simplify the equation to get the final answer. It comes out as \[ \frac{x^2}{16}+\frac{y^2}{49}=1 \].

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse equation helps you understand the basic structure of an ellipse.
An ellipse is a shape that looks like a stretched-out circle, and its equation reflects that. It's written as:
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
Here, \(a\) and \(b\) are values that help define the size of the ellipse. If \(a > b\), the ellipse is wider horizontally; if \(b > a\), it's taller vertically.
These values are squared in the equation, which is key for maintaining the shape's properties. Simplifying this equation can often help in solving problems related to ellipses. Remember, this form is applicable only when the center of the ellipse is at the origin, or \((0,0)\).

Center at the Origin

When an ellipse has its center at the origin, its equation becomes simpler to work with.
The origin is the point \((0,0)\) on a coordinate plane, where the x-axis and y-axis intersect.
Having the center at the origin means that neither \(x\) nor \(y\) in the equation of an ellipse will have any added constants (like \(x-h\) or \(y-k\)). Instead, it remains \(x^2\) and \(y^2\).
This makes it easier to identify and substitute the values of \(a\) and \(b\) directly into the standard form, streamlining the solving process.

Vertex of an Ellipse

The vertex of an ellipse is one of the most important points to understand.
It refers to the furthest points on the ellipse along its major axis.

• In a vertical ellipse, the vertices are at \( (0, -b) \) and \( (0, b) \).
• In a horizontal ellipse, they are at \( (-a, 0) \) and \( (a, 0) \).

Here, \(b\) is longer than \(a\) if the ellipse is vertical, or vice versa if horizontal.
The vertex helps determine the direction and the stretch of the ellipse. Especially when the center is at the origin, identifying the vertex gives critical insights into forming the correct equation.

Co-Vertex of an Ellipse

The co-vertex is another important feature of an ellipse, complementing the vertex.
These are the ends of the minor axis, the shorter axis when compared to the major axis.

• In a horizontal ellipse, the co-vertices are \((0, -b)\) and \((0, b)\).
• In a vertical ellipse, they are \((-a, 0)\) and \((a, 0)\).

Understanding the co-vertex offers essential perspective regarding the proportion and orientation of an ellipse.
Because an ellipse is symmetric, this axis helps maintain the balance in its shape across both the x and y axes.