Problem 1 - Analytical Geometry: 2D and 3D

Question

Find the equation of the cylinder, whose guiding curve is \(x^{2}+z^{2}-4 x-2 z+4=0\), \(y=0\) and whose axis contains the point \((0,3,0)\). Find also the area of the section of the cylinder by a plane parallel to the \(x z\) plane.

Step-by-Step Solution

Answer

Answer: The equation of the cylinder is (x-2)^2 + y^2 +(z-1)^2 = 1, and the area of its section cut by the plane is 9π.

Step 1: Find the equation of the cylinder

Given the guiding curve equation: \(x^2 + z^2 - 4x -2z + 4 = 0\) with \(y=0\). Let's rewrite the equation in standard form for a circle: \((x-a)^2 + (z-c)^2 = r^2\). To do this, we will complete the square in \(x\) and \(z\). First, find the coefficients for completing the square: \(a = \frac{4}{2} = 2\) \(c = \frac{2}{2} = 1\) Now, we need to find the additional constant term that would be added on both sides of the equation: \((x-2)^2 - 4 = x^2 - 4x + 4\) \((z-1)^2 - 1 = z^2 -2z + 1\) Substitute the expressions back into the equation of the guiding curve: \((x - 2)^2 - 4 + (z - 1)^2 - 1 + 4 = 0\) Now, the equation in the standard form: \((x-2)^2 +(z-1)^2 = 1\) The corresponding cylinder equation, without changing \(x\) and \(z\) components, is: \((x-2)^2 + y^2 +(z-1)^2 = 1\)

Step 2: Find the area of the section of the cylinder

Now we'll find the area of the section of the cylinder by a plane parallel to the \(xz\) plane. Since the plane is parallel to the \(xz\) plane, it will cut the cylinder to form a circle. The radius of this circle is equal to the magnitude of the \(y\)-coordinate of the point on the axis: \((0,3,0)\). Therefore, the radius of the circle is \(3\). Now we can find the area of the circle using the formula: Area = \(πr^2\) Where \(r = 3\), Area = \(π(3^2)\) Area = \(9π\) So, the area of the section of the cylinder by a plane parallel to the \(xz\) plane is \(9π\).

Key Concepts

Guiding Curve

Understanding the concept of a guiding curve is crucial when dealing with cylinder equations. The guiding curve serves as the 'base' around which the cylinder is formed. In the context of the provided exercise, the guiding curve is represented by the equation x^{2}+z^{2}-4x-2z+4=0, at y=0. This means that at the slice of the three-dimensional space where y equals zero, the points that satisfy this equation will form a curve—which is in this case a circle—that acts as the guide or path for the cylinder.

When the cylinder is extended in the y-direction, every cross-section parallel to the guiding curve is identical, forming the circular cylindrical structure. To visualize it, think of stacking coins, where each coin represents a guiding curve, and the stack represents the cylinder. This illustration helps in understanding that the cylinder is just a series of circles stacked along a straight line—the axis.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into a form that is easier to analyze and graph. This method is especially handy when dealing with the equations of conic sections like circles, ellipses, and parabolas. In our exercise, this technique is applied to restructure the guiding curve into the standard form of a circle's equation: (x-a)^2 + (z-c)^2 = r^2, where a and c represent the coordinates of the circle's center, and r is its radius.

As demonstrated in the solution steps, by completing the square for both x and z terms, we find that the center of the guiding curve's circle is at the point (2, 1) and its radius is 1. This modification simplifies the equation, making it more comprehensible and allowing us to further deduce the cylinder's equation by incorporating the unchanged y-component.

Cylinder Section Area

The cross-sectional area of a cylinder is a measurement of the surface area cut by a plane that is parallel to the base of the cylinder. For a right circular cylinder, like the one described in our exercise, when you cut through it parallel to the base, the resulting section will always be a circle. To find the area of this section, you can use the area formula for a circle, which is A = πr^2, where A is the area and r is the radius of the circle.

The problem states that the axis of the cylinder contains the point (0,3,0), which dictates that the radius of the cylinder is 3. Using the formula, the area of the cylinder's section is therefore 9π. This concept highlights how geometrical properties in three dimensions relate to traditional two-dimensional measurements, in this case linking the 3D cylinder to the 2D circle through their sections.