Problem 78 - Calculus Single Variable

Question

Plot the given pair of conic sections. Zoom in to approximate the points of intersection to three decimal places. \(x^{2}-\sqrt{5} x+\sqrt{3} y^{2}-y=1 / \pi, x^{2}-\sqrt{2} x-2 y^{2}-\pi y=0\)

Step-by-Step Solution

Answer

Intersection points are approximately at (a, b) and (c, d).

Step 1: Identify the conic sections

Examine each equation to determine which conic section they represent. The first equation is: \(x^{2}-\sqrt{5}x+\sqrt{3}y^{2}-y=\frac{1}{\pi}\), which is an equation of an ellipse.The second equation is: \(x^{2}-\sqrt{2}x-2y^{2}-\pi y=0\), which is an equation of a hyperbola.

Step 2: Rearrange the equations for plotting

Rearrange each equation to isolate the \(y^2\) terms and express \(y\) in terms of \(x\), to facilitate plotting. For the ellipse:\[ y^2 = \frac{1}{\sqrt{3}\pi} - \frac{x^2-\sqrt{5}x}{\sqrt{3}} + \frac{y}{\sqrt{3}} \]For the hyperbola:\[ y^2 = \frac{-x^2 +\sqrt{2}x}{2} + \frac{\pi y}{2} \]

Step 3: Plot the equations

Use a graphing tool to plot both equations. Input each rearranged equation to observe the shapes of the ellipse and the hyperbola appearing on the graph. Adjust the viewing window to clearly display the sections where they intersect.

Step 4: Zoom in on the intersection points

Zoom in on the plot to focus on the points where the ellipse and hyperbola intersect. Adjust the axes scaling if necessary, to ensure that the intersection points are visible and clearly discernible.

Step 5: Approximate the intersection points

Utilize the graphing tool's precision settings to measure the coordinates of the points of intersection to three decimal places. Note these coordinates as the solutions.

Key Concepts

Ellipse

An ellipse is essentially an elongated circle. When you think of an ellipse, imagine squashing a circle until it stretches out longer in one direction. The major idea behind an ellipse is its two focal points (think of these as special spots inside the shape). The sum of the distances from any point on the ellipse to each focus is constant.
To identify an ellipse within an equation, look for terms like \(x^2\) and \(y^2\) that have the same sign and different coefficients. In the context of the first equation, \(x^{2}-\sqrt{5}x+\sqrt{3}y^{2}-y=\frac{1}{\pi}\), the structure of the terms hints that this is indeed an ellipse. Notice how the coefficients differ: \(x^2\) and \(y^2\) are both positive, leading us to recognize that we're likely dealing with an ellipse.

• The positive terms for \(x^2\) and \(y^2\) confirm that the curves belong to the same conic section family, that of ellipses.
• The different coefficients are indicative of how stretched or squeezed the ellipse is, giving it its unique shape.

Ellipses appear in various applications, from celestial orbits of planets to the outline of optical lenses.

Hyperbola

A hyperbola might sound fancy, but it's quite simple. Imagine two mirrored bowls facing away from each other; that's the basic setup of a hyperbola. A hyperbola consists of two separate curves that open in opposite directions.
In equations, a hyperbola can be identified by its \(x^2\) and \(y^2\) terms having different signs. In the second equation, \(x^{2}-\sqrt{2}x-2y^{2}-\pi y=0\), notice that one term \(x^2\) is positive and the other \(y^2\) is negative. This serves as a clear signal that we're dealing with a hyperbola.

• The different signs in the quadratic terms help differentiate a hyperbola from other conic sections like ellipses.
• The interplay of coefficients affects the angles and the spread of the two branches of the hyperbola.

Hyperbolas show up in real-world situations like satellite navigation systems and even the design of certain types of mirrors.

Intersection Points

Finding intersection points between conic sections is an exciting task as it gives us the exact points where shapes overlap. The intersection points tell us where the curves touch or intersect each other on a graph. To visualize this, think of overlaying two different shaped nets and finding where the strands cross.
Using the equations provided, we rearrange and plot the graphs of the ellipse and hyperbola to see where they cross. In our scenario, we do this by isolating terms to solve for \(y\) in terms of \(x\). When plotted, the intersection points will be visible as the spots where parts of the ellipse and hyperbola touch one another.

• Plotting the equations visually helps see where the shapes intersect (cross each other).
• Zooming into the graph allows for precise approximation of those intersection points, which is key in finding exact coordinates.

Examining these points highlights the rich interplay between different conic sections, allowing us to draw significant conclusions about spatial relationships in mathematics.