Problem 45 - Precalculus Mathematics for Calculus

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 (0, 2) and (0, -2).

Step 1: Understand the Equations

We have two ellipse equations: \(4x^2 + y^2 = 4\) and \(4x^2 + 9y^2 = 36\). The goal is to find points \((x, y)\) that satisfy both equations simultaneously. With these equations, you can determine that one ellipse is vertical while the other is horizontal.

Step 2: Eliminate a Variable

To find the intersection points, express \(y^2\) from the first equation and substitute it into the second. For the first equation, solve for \(y^2\): \(y^2 = 4 - 4x^2\). Substitute into the second equation: \(4x^2 + 9(4 - 4x^2) = 36\).

Step 3: Simplify and Solve for \(x\)

Substitute \(y^2\) into \(4x^2 + 9y^2 = 36\): \[4x^2 + 36 - 36x^2 = 36\]. Simplify to get: \(-32x^2 = 0\). This implies \(x^2 = 0\), so \(x = 0\).

Step 4: Find \(y\) for \(x = 0\)

Substitute \(x = 0\) back into either equation. Using \(4x^2 + y^2 = 4\), substitute \(x = 0\), which leaves \(y^2 = 4\). Hence, \(y = \pm 2\).

Step 5: Determine Intersection Points

The intersection points are where both ellipses meet, which are \((0, 2)\) and \((0, -2)\).

Step 6: Sketch the Ellipses and Label Points

Sketch both ellipses on the same coordinate axes. The ellipse \(4x^2 + y^2 = 4\) intersects with the ellipse \(4x^2 + 9y^2 = 36\) at points \((0, 2)\) and \((0, -2)\). Ensure the points \((0, 2)\) and \((0, -2)\) are clearly marked as intersection points.

Key Concepts

Ellipse Equation

Ellipses are fascinating curves in geometry. They are defined by specific equations called ellipse equations. These equations determine the shape and orientation of the ellipse on a graph. An ellipse can be imagined as an elongated circle.

For any standard ellipse centered at the origin, you might encounter equations of forms like \ ax^2 + by^2 = c \. Here, \(a\), \(b\), and \(c\) are constants that affect the size and orientation of the ellipse:

• When \(a = b\), the ellipse becomes a circle.
• A larger value of \(a\) compared to \(b\) causes the ellipse to stretch more along the x-axis and vice versa.

In the exercise given, the equations are \(4x^2 + y^2 = 4\) and \(4x^2 + 9y^2 = 36\). Notice how these ellipses have different forms, which directly affect their stretches along the axes. The first ellipse stretches equally along both x and y axes, while the second ellipse stretches more along the y-axis due to the coefficient of \(9y^2\). This difference in coefficients dictates how the ellipses are drawn on the coordinate plane.

Coordinate Geometry

Coordinate geometry is essentially the use of algebraic methods to solve geometric problems. This branch of mathematics allows us to precisely locate points, lines, and shapes on a plane using coordinates.

In dealing with ellipses or any geometric shape, coordinate geometry helps you:

• Visualize the position and orientation of the figure by using coordinate axes.
• Determine intersection points of shapes by combining equations.
• Predict the shape changes by modifying the equation coefficients.

In the problem given, each ellipse is represented on the coordinate plane using its specific equation. By solving these equations algebraically, you can precisely find the intersection points, such as \(0, 2\) and \(0, -2\) for the current exercise. The power of coordinate geometry is in connecting abstract equations with visual, real-world representations.

Intersection Points

Finding intersection points between two shapes is a fundamental concept in geometry. This involves identifying shared points where the given curves meet on a graph. These points represent solutions that satisfy both equations.

To determine intersection points:

• Ensure the equations are simplified and if possible expressed in terms of one variable.
• Substitute or equate one equation into the other to find common solutions.
• Solve for one variable and use its value to find the corresponding values of the other variables.

In our example, we started by simplifying the equations \(4x^2 + y^2 = 4\) and \(4x^2 + 9y^2 = 36\). By substituting \(y^2\) from the first equation to the second, we managed to find that \(x^2 = 0\), so \(x = 0\). Using \(x = 0\), we calculated \(y = \pm 2\), giving us the intersection points \( (0, 2) \) and \( (0, -2) \). These points where both ellipses meet are crucial for understanding how these shapes interact on the graph.