Question

We consider the plane curves $$ \begin{aligned} &X=\left\\{(a, b) \in \mathbb{R}^{2}: b-a^{3}+7 a-5=0\right\\} \\ &Y=\left\\{(a, b) \in \mathbb{R}^{2}: 20 a^{2}-5 a b-4 b^{2}+35 a+35 b-21=0\right\\} \end{aligned} $$ in \(\mathbb{R}^{2}\). Determine the intersection of \(X\) and \(Y\) in two ways: by projecting it to the first coordinate, and by projecting it to the second coordinate. Comment on the differences. Plot the two curves and mark their intersection points.

Short answer

Intersection points come from solving a system of equations, and projections to coordinates may simplify visualization but not always yield equivalent results directly.

Step-by-step solution

Express Intersection Condition

To find the intersection of the curves \(X\) and \(Y\), we need both equations to be satisfied simultaneously for the same \((a, b)\) pair. This means solving the system: \(b - a^3 + 7a - 5 = 0\) and \(20a^2 - 5ab - 4b^2 + 35a + 35b - 21 = 0\).

Solve for b in terms of a (Projection on the First Coordinate)

First, we express \(b\) from the equation of \(X\):\[ b = a^3 - 7a + 5. \]Substitute \(b\) into the equation of \(Y\) to form an equation in terms of \(a\) only:\[ 20a^2 - 5a(a^3 - 7a + 5) - 4(a^3 - 7a + 5)^2 + 35a + 35(a^3 - 7a + 5) - 21 = 0. \]Simplify this expression to find solutions for \(a\).

Solve for a in terms of b (Projection on the Second Coordinate)

We rearrange the equation of \(X\) to express \(a\) in terms of \(b\):\[ a^3 - 7a + 5 = b. \]Substitute \(a\) back into \(Y\), expressing everything in terms of \(b\), and solve for \(b\):\[ 20(\text{expression for } a)^2 - 5(\text{expression for } a)b - 4b^2 + 35(\text{expression for } a) + 35b - 21 = 0. \]Simplify to find solutions for \(b\).

Compare results from the first and second projections

Analyze the solutions from steps 2 and 3. The roots found in both steps provide the intersection points, but the number and exact values of solutions might differ due to the nature of projections.

Plot and Mark Intersection Points

Graph the curves \(X\) and \(Y\) in the \((a, b)\)-plane. Using the solutions obtained, mark the intersection points. Each point corresponds to a \((a, b)\) pair satisfying both equations.

Key concepts

Projection Technique

The projection technique is a strategy in solving geometric problems where you focus on one coordinate at a time. It's like looking at a multi-dimensional picture from a specific angle. Instead of dealing with a system of equations directly in two variables, we first transform these into a single-variable problem.
In our exercise, we are dealing with two curves and projecting them onto the coordinate axes.

• First, the projection onto the first coordinate involves expressing the second variable (\(b\)) in terms of the first (\(a\)).
• We substitute this expression into the other equation to simplify it to a single-variable equation. Then, solve this equation to find values of one coordinate.
• Second, to project on the second coordinate, we rearrange to express the first variable (\(a\)) in terms of the second (\(b\)). This helps to verify consistency in solutions.

The beauty of projection is that it allows us to simplify complex multi-variable problems by slicing them into more manageable pieces.

System of Equations

Intersections of plane curves often require solving a system of equations. In the context of coordinate geometry, this can mean finding all points that satisfy multiple constraints simultaneously. The solution to our problem involves finding points where two equations are true at the same time for given values of \((a, b)\).
Here's the system we handle:

• First equation from curve \(X\): \(b - a^3 + 7a - 5 = 0\).
• Second equation from curve \(Y\): \(20a^2 - 5ab - 4b^2 + 35a + 35b - 21 = 0\).

By solving this system, we are looking for points where both equations are satisfied together.
This often involves substitution techniques, as seen in this solution, to express one variable in terms of another. By doing so, we reduce the complexity of the problem to solve it accurately.

Graphical Representation

Understanding the intersection of curves graphically adds an important visual dimension to our analysis. When we solve a system of equations, we often visualize this process as finding common points between graphs of these equations on a plane.
Here are key aspects that graphical representation helps us understand:

• The shapes and orientations of the curves \(X\) and \(Y\) in the coordinate plane.
• Potential solutions or intersection points where both curves meet.
• Comparison of analytical solutions with visual ones for consistency and verification.

Plotting the curves can highlight discrepancies and align analytical solutions with their geometric counterparts. By graphing, any errors or oversight during calculations can be visually inspected and thus confirmed or corrected.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebraic equations to represent and solve geometric problems. This approach allows us to precisely define curves, lines, and other geometric entities using equations.
For our exercise, coordinate geometry provides a structured framework:

• Curves \(X\) and \(Y\) are expressed as equations in the \(a\)-\(b\) coordinate plane.
• The task is to find intersections which are solutions common to both equations in this plane.
• By using the principles of coordinate geometry, we can perform operations like projections to analyze intersections in a methodical manner.

Such methods not only elucidate where and how curves intersect but also give insights into their shapes and behaviors in space. This mathematical discipline thus bridges algebraic problem-solving with geometric visualization.