Problem 80 - Precalculus: Functions and Graphs

Question

Graph the hyperbolas on the same coordinate plane, and estimate their first- quadrant point of intersection.. $$\frac{(x-0.1)^{2}}{0.12}-\frac{y^{2}}{0.1}=1 ; \frac{x^{2}}{0.9}-\frac{(y-0.3)^{2}}{2.1}=1$$

Step-by-Step Solution

Answer

Graphs visually show intersections; intersection point estimated near (0.3, 0.0).

Step 1: Identify the type of conic sections

We recognize that the given equations are hyperbolas because they are in the standard form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \). In each given equation, the first term involves a subtraction and they fit this structure, indicating hyperbolas.

Step 2: Determine the foci and vertices of first hyperbola

The first equation is \( \frac{(x-0.1)^2}{0.12} - \frac{y^2}{0.1} = 1 \). This hyperbola is centered at \( (0.1, 0) \), with semi-major axis \( \sqrt{0.12} \) and semi-minor axis \( \sqrt{0.1} \). The vertices are at \( (0.1 \pm \sqrt{0.12}, 0) \) and foci can be calculate using \( c = \sqrt{a^2 + b^2} \.\)

Step 3: Determine the foci and vertices of second hyperbola

The second equation is \( \frac{x^2}{0.9} - \frac{(y-0.3)^2}{2.1} = 1 \). This hyperbola is centered at \( (0, 0.3) \), with semi-major axis \( \sqrt{0.9} \) and semi-minor axis \( \sqrt{2.1} \). The vertices are at \( (\pm \sqrt{0.9}, 0.3) \) and foci can be calculate using \( c = \sqrt{a^2 + b^2} \.\)

Step 4: Graph the hyperbolas

Plot the hyperbolas using their center, vertices, and a rough outline provided by the semi-major and semi-minor axes lengths. Ensure both are on the same set of axes for comparison.

Step 5: Find the intersection points algebraically

Substitute \( y = 0 \) in both hyperbola equations given their orientations do not favor immediate graphical intersection in the first quadrant. Solve \(-\) for real solutions to find the intersection by equating expressions.

Step 6: Estimate intersection in first quadrant

Examine the solutions graphically to see where the hyperbolas intersect in the graph's first quadrant, which is where both variables are positive.

Key Concepts

Conic Sections

Conic sections are fascinating curves that form when a plane intersects with a double-napped cone. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each of these has its own unique equation and graphical representation.

A hyperbola is one of these conic sections that can be identified by its equation of the form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \). It is distinct because it contains a subtraction between the squared terms, setting it apart from circles and ellipses which use addition. A hyperbola consists of two separate curves, or branches, which tend to open away from each other.

The center of a hyperbola is given by the point \((h, k)\), and the axes lengths are derived from \(a\) and \(b\). These parameters help locate the hyperbola's vertices and define its shape. Understanding the foundation of these equations is key when identifying conic sections in geometry problems. By recognizing this, students can quickly determine if an equation represents a hyperbola and proceed with graphing and analyzing it.

Graphing Hyperbolas

Graphing a hyperbola requires understanding its structure based on the standard form equation. Once you identify a hyperbola's equation, the next step is to locate its center, which acts as a reference point for plotting.

The vertices of a hyperbola are crucial. They indicate where the hyperbola takes its widest point in each direction. For an equation in the form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), the vertices lie at \((h \pm a, k)\) or \((h, k \pm b)\), depending on the orientation—horizontal or vertical.

• Calculate the semi-major axis length \(a\) by taking the square root of the denominator under the x-term.
• Calculate the semi-minor axis length \(b\) by taking the square root of the denominator under the y-term.
• Locate the foci using \( c = \sqrt{a^2 + b^2} \), which provides the distance from the center to each focus along the same axis as the vertices.

After determining these properties, plot the center, vertices, and foci. Then, sketch the hyperbola by smoothly connecting these points. Repeat this for each hyperbola given in a problem, ensuring that they are plotted on the same coordinate plane for comparison, as seen in our example.

Intersection Points

Finding intersection points of hyperbolas involves analyzing where the curves meet in a given plane. For this purpose, algebraic techniques often come into play to determine the precise coordinates.

When two hyperbolas are plotted on the same set of axes, potential intersection points are where the equations of both hyperbolas hold true simultaneously. The first step to solve such a problem is to simplify the equations if possible, or to substitute variables that might help find common solutions. For instance, setting one variable (like \( y = 0 \)) can simplify equations if it is suitable based on the known orientations of the hyperbolas.

• Solve these simplified equations to find any real solutions.
• Evaluate the solutions to determine their validity in the specific quadrant being examined, such as the first quadrant where both \( x \) and \( y \) are positive.

Intersection points can also be estimated graphically by examining plot points where hyperbolas visibly intersect. This approach aids in verifying the algebraic solutions and ensures a complete understanding of where and why these interactions occur within the graph.