Question

Show that one can solve \(x^{3}+d=c x\) by intersecting the hyperbola \(y^{2}-x^{2}+\frac{d}{c} x=0\) with the parabola \(x^{2}=\sqrt{c} y\). Sketch the two conics. Find sets of values for \(c\) and \(d\) for which these conics do not intersect, intersect once, and intersect twice.

Short answer

Question: Show that one can solve the cubic equation \(x^3+d=cx\) by finding the intersection of the hyperbola \(y^2-x^2+\frac{d}{c}x=0\) and the parabola \(x^2=\sqrt{c}y\). Answer: We demonstrated that the cubic equation is equivalent to finding the intersection of the hyperbola and the parabola by rewriting the equation in terms of these conics. To solve the cubic equation, we determined conditions on \(c\) and \(d\) for various intersection scenarios: - No intersection: \(c < 4\) and \(d\) large enough such that the hyperbola does not intersect the \(x\)-axis. - One intersection: \(d = 0\). - Two intersections: \(c > 4\) and \(d > 0\).

Step-by-step solution

Rewrite the cubic equation

We'll start by subtracting \(cx\) from both sides to obtain \(x^3 - cx + d = 0\). Next, we'll introduce a new variable \(y\) by adding \(x^2\) to both sides, which results in: \(x^3 - cx + d + x^2 = x^2\). Now, let's rewrite the left side of our equation as the difference of squares: \(x^3 - cx + d + x^2 = (y^2 - x^2) + x^2 + \frac{d}{c}x + \frac{c-d}{c}x\). Matching the terms, we get: \(y^2 - x^2 = -x^2\), \(x^2 = \sqrt{c}y\), and \(\frac{d}{c}x = \frac{c-d}{c}x\). So, the cubic equation is equivalent to finding the intersection of the hyperbola \(y^2 - x^2 + \frac{d}{c}x = 0\) and the parabola \(x^2 = \sqrt{c}y\).

Sketch the conics

Now that we have found the conics, let's sketch them. The hyperbola \(y^2 - x^2 + \frac{d}{c}x = 0\) has a center at \((0, 0)\) and its main axis is along the \(x\)-axis. It opens left and right, its asymptotes are the \(45^\circ\) and \(135^\circ\) lines. The parabola \(x^2 = \sqrt{c}y\) has the vertex \((0, 0)\) and opens upward. The focal length is \(\frac{1}{4\sqrt{c}}\), and thus the focus is at the point \((0, \frac{1}{4\sqrt{c}})\).

Determine conditions on \(c\) and \(d\) for number of intersections

To determine the intersection conditions for \(c\) and \(d\), we'll analyze the relative positions of the conics. 1. No intersection: This would happen when the vertex of the parabola lies inside the hyperbola, or if the parabola is too narrow to intersect the hyperbola. For the first case, the focal length must be greater than the distance from the focus to the asymptote of the hyperbola, which is \(\frac{1}{2}\). So, \(\frac{1}{4\sqrt{c}} > \frac{1}{2}\), which implies \(c < 4\). For the second case, \(d\) must be large enough such that the hyperbola does not intersect the \(x\)-axis. 2. One intersection: This occurs when the vertex of the parabola lies on the hyperbola, so: \(0 - 0 + \frac{d}{c} \cdot 0 = 0 \Rightarrow d = 0\). In this case, the parabola is tangent to the hyperbola. 3. Two intersections: This happens when the parabola and hyperbola intersect in two distinct places. This requires the vertex of the parabola to lie outside the hyperbola, or another way to say this, the focal length is smaller than the distance from the focus to the asymptote of the hyperbola. So, \(\frac{1}{4\sqrt{c}} < \frac{1}{2}\), which implies \(c > 4\). We also need the hyperbola to intersect the \(x\)-axis, which means \(\frac{d}{c}x > 0\) for \(x > 0\), or \(d > 0\) since \(c > 0\). Based on this analysis, we can conclude the following: - No intersection: \(c < 4\) and \(d\) large enough such that the hyperbola does not intersect the \(x\)-axis. - One intersection: \(d = 0\). - Two intersections: \(c > 4\) and \(d > 0\).

Key concepts

Cubic Equations

Cubic equations have the general form \(ax^3 + bx^2 + cx + d = 0\). They are algebraic equations of degree three, meaning the highest power of the variable \(x\) is three. These equations can have up to three roots, including real or complex numbers, depending on the coefficients \(a, b, c,\) and \(d\). For example, the equation \(x^3 - cx + d = 0\) is a cubic equation with terms for \(x^3, x\) and a constant term \(d\).

Solving a cubic equation can be more intricate than solving linear or quadratic equations. Sometimes, techniques like factoring using the sum or difference of cubes, synthetic division, or the use of Cardano's formula are applied. However, in certain cases, cubic equations can also be visualized through graphing, using geometrical interpretations like intersections of curves, providing additional insights into their solutions.

Intersection of Curves

The intersection of curves concept is fundamental in graphing and solving equations through graphical representations. When two curves intersect, their intersection points represent the solutions common to both curves. In the context of the exercise, the intersection points of the hyperbola \(y^2 - x^2 + \frac{d}{c} x = 0\) and the parabola \(x^2 = \sqrt{c}y\) give solutions to the cubic equation \(x^3 + d = cx\).

Finding these intersection points involves setting the equations of the curves equal and resolving for the common coordinates \((x, y)\). Such graphical solutions provide an intuitive way to understand how the solutions exist, however many they might be. Notably, the conditions for intersection can range from zero, one, or two real intersections, based on the parameters \(c\) and \(d\), affecting the hyperbola and parabola's shape and position.

Analytic Geometry

Analytic geometry, also known as coordinate or Cartesian geometry, is a branch of mathematics that uses a coordinate system to represent and solve problems involving geometric figures. It forms a bridge between algebraic equations and geometric curves, allowing for the interpretation of algebraic equations as geometric shapes.

The problem uses analytic geometry to describe conic sections such as hyperbolas and parabolas through equations involving \(x\) and \(y\). These equations provide a visual and analytical way to solve algebraic problems like cubic equations by considering their graphs in a coordinate plane.

• Equations of the conics are manipulated algebraically to find intersections.
• Translating these into geometric curves, intersections correspond to solutions of the algebraic equations.
• Analytic geometry techniques help in understanding curves' properties and relationships.

Graphing Conic Sections

Graphing conic sections involves plotting the different types of curves that can be obtained by intersecting a plane with a cone: circles, ellipses, parabolas, and hyperbolas. Each conic section has a unique equation and represents different geometric properties.

In the problem, we deal with a hyperbola \(y^2 - x^2 + \frac{d}{c} x = 0\) and a parabola \(x^2 = \sqrt{c}y\). Understanding how to graph these conics is crucial as it helps to visually determine their points of intersection.

Graphing requires recognizing the standard forms of the conic sections and interpreting transformations like shifts and stretches due to parameters. When drawn, we can observe their orientation, axes of symmetry, and key features such as vertices and foci, crucial for understanding the overall behavior of intersections. This approach makes algebraic problems more tangible and often easier to solve through an intuitive graphical perspective.