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Chapter 12: Problem 49

Graph each ellipse. $$\frac{y^{2}}{25}=1-\frac{x^{2}}{49}$$

Short Answer

Expert verified
Standard form: \(\frac{y^2}{25} + \frac{x^2}{49} = 1\), vertices: (7, 0) and (-7, 0), co-vertices: (0, 5) and (0, -5).

Step by step solution

01

Rewrite the equation in standard form

Given equation: \(\frac{y^2}{25} = 1 - \frac{x^2}{49}\). To rewrite this in standard form, add \(\frac{x^2}{49}\) to both sides: \ \(\frac{y^2}{25} + \frac{x^2}{49} = 1\). This is the standard form of an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
02

Identify the values of a and b

From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), it can be identified that \(a^2 = 49\) and \(b^2 = 25\). Thus, \(a = 7\) and \(b = 5\).
03

Identify the center, vertices, and co-vertices

The center of the ellipse is at the origin (0, 0). The vertices are at \((\text{卤}a, 0)\), which gives points (7, 0) and (-7, 0). The co-vertices are at \((0, \text{卤}b)\), which gives points (0, 5) and (0, -5).
04

Graph the ellipse

To graph the ellipse, plot the center at (0, 0), the vertices at (7, 0) and (-7, 0), and the co-vertices at (0, 5) and (0, -5). Draw a smooth curve connecting these points to form the ellipse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of Ellipse
An ellipse is a geometric shape that looks like a stretched circle. The standard form of the equation of an ellipse is useful because it allows us to easily find important characteristics like its center, vertices, and co-vertices.
The standard form of an ellipse is: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) Here,
  • a is the length of the semi-major axis (half the longest diameter)
  • b is the length of the semi-minor axis (half the shortest diameter)
In the exercise's equation, \(\frac{y^2}{25} = 1 - \frac{x^2}{49}\), we need to rewrite it in the standard form:
  • Add \(\frac{x^2}{49}\) to both sides: \(\frac{y^2}{25} + \frac{x^2}{49} = 1\)
  • Now it matches the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a^2 = 49\) and \(b^2 = 25\)
Knowing the equation in standard form helps us easily find the ellipse's parts.
Vertices and Co-vertices
Vertices and co-vertices are specific points located on the ellipse. They help define the size and shape of the ellipse.
Let's identify them for our exercise:
  • **Vertices**: These points are located at the end of the major axis (longest diameter). For our ellipse, \(a = 7\). So, the vertices would be at \( (\text{卤}a, 0) \), which are (7, 0) and (-7, 0).
  • **Co-vertices**: These are located at the end of the minor axis (shortest diameter). For our ellipse, \(b = 5\). So, the co-vertices would be at \( (0, \text{卤}b) \), which are (0, 5) and (0, -5).
Plotting these points on a graph helps us understand the overall size and exact shape of the ellipse.
Center of Ellipse
The center of an ellipse is the midpoint from which the axes (major and minor) extend. It is an important feature because it helps in plotting and understanding the symmetry of the ellipse.
For the given exercise, the equation of the ellipse does not contain \(h\) and \(k\) terms (like \( (x - h)^2 \) or \( (y - k)^2 \)). This means the center is at the origin \( (0, 0) \).
Let's visualize how to plot:
  • Place a dot at the center (0, 0).
  • Mark the vertices at (7, 0) and (-7, 0).
  • Mark the co-vertices at (0, 5) and (0, -5).
  • Draw a smooth curve through these points to form the ellipse.
Understanding the center helps us in shifting the graph of the ellipse if needed, by adjusting \(h\) and \(k\) in the equation \((x - h)^2/a^2 + (y - k)^2/b^2 = 1\).

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