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Chapter 12: Problem 20

Identify the graph of each equation as a parabola, circle, ellipse, or hyperbola, and then sketch the graph. $$ 9 x^{2}+25 y^{2}=225 $$

Short Answer

Expert verified
The equation represents an ellipse. The semi-major axis is 5, and the semi-minor axis is 3. Sketch the ellipse centered at the origin with the points (5,0), (-5,0), (0,3), and (0,-3).

Step by step solution

01

- Identify the Type of Equation

Compare the given equation to the standard forms of conic sections. The given equation is: \[ 9x^{2} + 25y^{2} = 225 \]If an equation is in the form \[ Ax^{2} + By^{2} = C \] where A and B are positive and different, it is the equation of an ellipse.
02

- Rewrite the Equation in Standard Form

Divide each term by the constant term on the right side of the equation to normalize it. This helps identify A and B more clearly.\[ \frac{9x^{2}}{225} + \frac{25y^{2}}{225} = \frac{225}{225} \]Simplifying, we get\[ \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 \]
03

- Identify Parameters of the Ellipse

The standard form of an ellipse is \[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \] where a and b are the semi-major and semi-minor axes respectively. Comparing the equation \[ \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 \] with the standard form, it's clear that\[ a^{2} = 25 \] and \[ b^{2} = 9 \]Therefore, \[ a = 5 \] and \[ b = 3 \]. This implies the ellipse has a semi-major axis of 5 along the x-axis and a semi-minor axis of 3 along the y-axis.
04

- Sketch the Graph

Draw the axes and plot the lengths of the semi-major and semi-minor axes:1. Center at the origin (0, 0).2. Along the x-axis, plot points at (5, 0) and (-5, 0) representing the endpoints of the major axis.3. Along the y-axis, plot points at (0, 3) and (0, -3) representing the endpoints of the minor axis.4. Connect these points in an elliptical shape.

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Key Concepts

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

Understanding the Ellipse Equation
An ellipse is a fascinating shape in geometry. Its equation typically looks like this: \( Ax^{2} + By^{2} = C \). In the exercise, we were given an equation: \( 9x^{2} + 25y^{2} = 225 \). To determine if this is an ellipse, we compare it to the standard form \( Ax^{2} + By^{2} = C \), where both A and B are positive but different. Here:
  • A = 9
  • B = 25
Identifying the Semi-Major Axis
To find the semi-major axis of an ellipse, first, we need to rewrite the equation in a more familiar form. We did this by normalizing it: \( \frac{9x^{2}}{225} + \frac{25y^{2}}{225} = 1 \) Now, it simplifies to \( \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 \). In the standard form of an ellipse, \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), 'a' represents the semi-major axis. Here
  • \( a^{2} = 25 \) so \( a = 5 \)
The semi-major axis is the longer one, and it lies along the x-axis. In this case, it stretches from -5 to 5 on the x-axis.
Identifying the Semi-Minor Axis
The semi-minor axis is the shorter one in an ellipse. Using the simplified equation: \( \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 \), we find 'b'. In the standard form of an ellipse, \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), 'b' represents the semi-minor axis. Here,
  • \( b^{2} = 9 \) so \( b = 3 \)
The semi-minor axis lies along the y-axis, stretching from -3 to 3. Remember:
  • Semi-major axis is the longer one (a = 5)
  • Semi-minor axis is the shorter one (b = 3)
Standard Form of Ellipse
The standard form of an ellipse equation is crucial in understanding its shape and orientation. The standard form is: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) where 'a' and 'b' are the lengths of the semi-major and semi-minor axes respectively. This form tells us:
  • How 'stretched' the ellipse is along the x and y axes.
  • Whether the ellipse is wider along the x or y-axis.
In our example, we turned \( 9x^{2} + 25y^{2} = 225 \) into \( \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 \). This revealed that the ellipse is wider along the x-axis with the following properties:
  • 'a' (semi-major axis) is along the x-axis
  • 'b' (semi-minor axis) is along the y-axis

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