Problem 45 - Algebra and Trigonometry

Question

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. $$\left\\{\begin{array}{l}{4 x^{2}+y^{2}=4} \\ {4 x^{2}+9 y^{2}=36}\end{array}\right.$$

Step-by-Step Solution

Answer

The ellipses intersect at the points (0, 2) and (0, -2).

Step 1: Write the equations in standard form

Rearrange both equations to be in the form of an ellipse: For the first equation, divide everything by 4: \( \frac{x^2}{1} + \frac{y^2}{4} = 1 \). For the second equation, divide everything by 36: \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \).

Step 2: Identify the ellipses

The first equation \( \frac{x^2}{1} + \frac{y^2}{4} = 1 \) represents an ellipse centered at the origin with a semi-major axis of 2 along the y-axis and a semi-minor axis of 1 along the x-axis. The second equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) represents an ellipse centered at the origin with a semi-major axis of 3 along the x-axis and a semi-minor axis of 2 along the y-axis.

Step 3: Equate the equations to find intersection points

Set \( 4x^2 + y^2 = 4 \) equal to \( 4x^2 + 9y^2 = 36 \) to find the intersection points. Simplify by subtracting the first equation from the second: \( (4x^2 + 9y^2) - (4x^2 + y^2) = 36 - 4 \), resulting in \( 8y^2 = 32 \).

Step 4: Solve for y

Divide both sides of \( 8y^2 = 32 \) by 8: \( y^2 = 4 \). Taking the square root of both sides yields two solutions: \( y = 2 \) and \( y = -2 \).

Step 5: Substitute back to find x-coordinates

Substitute \( y = 2 \) into the first equation \( 4x^2 + y^2 = 4 \): \( 4x^2 + 4 = 4 \) which simplifies to \( 4x^2 = 0 \). Thus, \( x^2 = 0 \) and \( x = 0 \). The same result occurs when substituting \( y = -2 \).

Step 6: List the intersection points

The points of intersection are \((0, 2)\) and \((0, -2)\).

Step 7: Sketch the graphs and label points

Draw the two ellipses on the same coordinate plane. The first ellipse is taller with its major axis on the y-axis, and the second ellipse is wider with its major axis on the x-axis. Plot and label the intersection points \((0, 2)\) and \((0, -2)\).

Key Concepts

Understanding Ellipse Equations

An ellipse is a set of points where the sum of the distances from two fixed points, called foci, is constant. To describe an ellipse mathematically, we use an ellipse equation. Each ellipse is defined by its own specific equation. In the exercise, we start with two equations:

• First: \( 4x^2 + y^2 = 4 \)
• Second: \( 4x^2 + 9y^2 = 36 \)

These equations help in identifying the shape, size, and orientation of the ellipse on a coordinate plane. Breaking down and manipulating these equations allows us to gain deeper insights into their geometric properties and find intersection points.

Exploring the Standard Form of Ellipse

The standard form of an ellipse's equation is given as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) represent the lengths of the semi-major and semi-minor axes, respectively. Converting an ellipse's equation into this standard form facilitates easier graphing and analysis. For the given exercise:

• The first equation, \( \frac{x^2}{1} + \frac{y^2}{4} = 1 \) indicates an ellipse with a semi-major axis length of 2 along the y-axis and a semi-minor axis length of 1 along the x-axis.
• The second equation, \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), represents an ellipse with a semi-major axis length of 3 along the x-axis and a semi-minor axis length of 2 along the y-axis.

These transformations assist in visualizing the ellipses more clearly and determining where they intersect.

The Process of Solving Simultaneous Equations

Finding the intersection of ellipses involves solving their equations simultaneously. By equating and manipulating these equations, we can find common points they share. In our exercise:

• We equate the two original equations: \( 4x^2 + y^2 = 4 \) and \( 4x^2 + 9y^2 = 36 \).
• Subtract the first equation from the second to eliminate \( x^2 \) terms, resulting in \( 8y^2 = 32 \).
• This simplifies further to \( y^2 = 4 \), yielding \( y = 2 \) and \( y = -2 \).

By solving these equations, we identify potential y-values of intersection. Substituting back into one of the original equations, we find the corresponding x-values. This approach allows for finding precise points of intersection, like \((0,2)\) and \((0,-2)\) in this case.

Graphing Ellipses and Identifying Intersections

Graphing provides a visual method to better understand the behavior and interaction of ellipses on a coordinate plane. To graph:

• Plot each ellipse separately, using their standard forms: the first is taller with a major axis along the y-axis, while the second is wider with its major axis along the x-axis.
• Mark the center at the origin \((0,0)\) for both ellipses.
• Identify where the paths of the ellipses intersect based on the calculated points \((0, 2)\) and \((0, -2)\).

This visualization not only confirms the solution found algebraically but also equips students with a stronger intuitive grasp of how ellipses intersect on the plane. It's a complementary tool that bolsters understanding of geometric relationships.