Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$ \left\\{\begin{array}{l} x^{2}-y^{2}=4 \\ y^{2}-3 x=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The points of intersection are (4, 2√3) and (4, -2√3).

Step 1: Understand the Given Equations

We are given two equations: 1. \( x^2 - y^2 = 4 \)2. \( y^2 - 3x = 0 \)Our task is to find their points of intersection by solving this system simultaneously.

Step 2: Solve the Second Equation for y

From the equation \( y^2 - 3x = 0 \), substitute to express \( y^2 \) in terms of \( x \): \[ y^2 = 3x \]

Step 3: Substitute into the First Equation

Substitute \( y^2 = 3x \) from Step 2 into the first equation \( x^2 - y^2 = 4 \): \[ x^2 - 3x = 4 \]

Step 4: Rearrange and Solve for x

Rearrange the equation \( x^2 - 3x = 4 \) to solve for \( x \): \[ x^2 - 3x - 4 = 0 \]Factor the quadratic equation:\[ (x - 4)(x + 1) = 0 \] Thus, \( x = 4 \) or \( x = -1 \).

Step 5: Solve for y with each x-value

For \( x = 4 \), substitute into \( y^2 = 3x \):\[ y^2 = 12 \] \( y = \pm \sqrt{12} = \pm 2\sqrt{3} \).For \( x = -1 \), substitute into \( y^2 = 3x \):\( y^2 = -3 \). Since \( y^2 \) cannot be negative in real numbers, there are no real solutions for \( x = -1 \).

Step 6: Summary of Points of Intersection

The points of intersection for the system are: 1. \( (4, 2\sqrt{3}) \)2. \( (4, -2\sqrt{3}) \)These are the points where both graphs intersect.

Key Concepts

Quadratic Equations

Quadratic equations play a key role in algebra and geometry, often appearing in a variety of mathematical problems. A quadratic equation is defined generally as an equation of the form \( ax^2 + bx + c = 0 \), where \( a eq 0 \). The solutions to a quadratic equation are known as the roots. These roots can be found using various techniques such as factoring, completing the square, or utilizing the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In our exercise, the given equation \( x^2 - 3x - 4 = 0 \) is already a quadratic form. By factoring, we found the roots to be \( x = 4 \) and \( x = -1 \). Understanding roots is crucial as they allow us to locate where the function, or graph, equals zero along the x-axis. In the context of intersection, solving quadratic equations helps determine the x-coordinates where two graphs meet.

Graphical Intersection

Graphical intersection refers to the point or points where two or more graphs meet on a coordinate plane. Identifying intersections is a valuable tool for visualizing solutions to a system of equations. These intersection points represent solutions where both equations hold true simultaneously. In our problem, we find the intersection points of two graphs, represented by the equations \( x^2 - y^2 = 4 \) and \( y^2 - 3x = 0 \). By substituting one equation into the other, we reduced the system to a single quadratic equation. Solving this gave us the potential x-values of the intersection points. Calculating the corresponding y-values revealed the exact points where the graphs intersect on the coordinate plane – the points \((4, 2\sqrt{3})\) and \((4, -2\sqrt{3})\). These intersections are crucial for understanding the relationships between different functions and their graphical representations.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is a branch of geometry where algebra is used to study geometric problems through a coordinate system. It provides a connection between algebraic equations and geometric figures, allowing for precise calculations of distances, midpoints, slopes, and intersection points.In this exercise, we used coordinate geometry to map the relationship between the algebraic system of equations and their geometric representation on a Cartesian plane. The equations \( x^2 - y^2 = 4 \) and \( y^2 - 3x = 0 \) describe curves on the plane. Solving the system reveals how these curves intersect, providing insight into their position and nature. Understanding the principles of coordinate geometry is essential for effectively interpreting and solving problems involving line and curve graphs in space.