Problem 19 - The Calculus with Analytic Geometry

Question

In Exercises 7 through 28 , draw a sketch of the graph of the equation. $$ 4 x^{2}+9 y^{2}=36 $$

Step-by-Step Solution

Answer

The graph is an ellipse centered at the origin with vertices at \(3, 0\) and \(-3, 0\), and co-vertices at \((0, 2)\) and \((0, -2)\).

Step 1: Identify the Type of Conic Section

The given equation is in the form of \(Ax^2 + By^2 = C\). Since both x and y terms are squared with positive coefficients, the equation represents an ellipse.

Step 2: Rewrite the Equation in Standard Form

Divide the entire equation by 36 to get the standard form of the ellipse: \[ \frac{4x^2}{36} + \frac{9y^2}{36} = 1 \]. Simplify it to: \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

Step 3: Identify the Lengths of the Axes

From the standard form \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \], we identify that \(a^2 = 9 \) and \(b^2 = 4 \). Hence, the lengths of the semi-major axis \(a = 3 \) and the semi-minor axis \(b = 2 \).

Step 4: Plot the Center

The center of the ellipse is at the origin \((0, 0)\).

Step 5: Plot the Vertices

The vertices of the ellipse are located on the x-axis, \((\pm a, 0)\). So, plot the vertices at \((3, 0)\) and \((-3, 0)\).

Step 6: Plot the Co-Vertices

The co-vertices are located on the y-axis, \((0, \pm b)\). So, plot the co-vertices at \((0, 2)\) and \((0, -2)\).

Step 7: Draw the Ellipse

Draw a smooth curve connecting the vertices and co-vertices to form the ellipse.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These shapes are parabolas, hyperbolas, circles, and ellipses.
Ellipses are among the most commonly encountered conic sections. Recognize them by their characteristic oval shape.
If an equation has both squared x and y terms and both coefficients are positive, it's usually an ellipse. For example, the equation provided, \[ 4x^{2}+9y^{2}=36 \], is an ellipse.

Standard Form of Ellipse

The standard form of an ellipse's equation makes it easier to understand and graph. The general form is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]. In our example, we need to divide by the constant on the right-hand side:
\[ \frac{4x^2}{36} + \frac{9y^2}{36} = 1 \]
Simplifying gives us:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
Now the equation is in its standard form, enabling us to identify key characteristics.

Axis Lengths

From the standard form \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \], we see that \[ a^2 = 9 \] and \[ b^2 = 4 \].
Therefore,
\[ a = 3 \] and \[ b = 2 \].
These values represent the lengths of the semi-major and semi-minor axes. The semi-major axis is the longest, running along the x-axis with a length of 6. The semi-minor axis is shorter, running along the y-axis with a length of 4.

Vertices and Co-Vertices

Vertices and co-vertices are crucial in understanding the shape of an ellipse. The vertices lie along the major axis, defined as \[ (\text{±}a, 0) \]. For our ellipse, the vertices are at
\[ (3,0) \] and \[ (-3,0) \].
Co-vertices lie along the minor axis, at
\[ (0,\text{±}b) \]. That means the co-vertices for our example are at
\[ (0, 2) \] and \[ (0, -2) \].
These points help us shape and position the ellipse accurately.

Plotting Graphs

Plotting the ellipse is straightforward when you have all key points. Start by marking the center at the origin \[ (0, 0) \].
Next, plot the vertices \[ (3, 0) \] and \[ (-3, 0) \] along the x-axis.
Then, plot the co-vertices \[ (0, 2) \] and \[ (0, -2) \] along the y-axis.
Finally, draw a smooth curve connecting these points to form the ellipse.
Always ensure the curve is as symmetrical as possible, following the shape indicated by the vertices and co-vertices.