Problem 61 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the points of \(x^{2}+14 x y+49 y^{2}=100\) that are closest to the origin.

Step-by-Step Solution

Answer

The closest points to the origin are (10,0) and (-10,0).

Step 1: Identify the curve

The given equation, \(x^2 + 14xy + 49y^2 = 100\), represents a conic section. To determine its type, we will compare it to a standard conic equation and transform it if needed.

Step 2: Rewrite the equation as a circle

Notice that \((x + 7y)^2 = x^2 + 14xy + 49y^2\). Thus, the equation can be rewritten as \((x + 7y)^2 = 100\). This represents two parallel lines: \(x + 7y = 10\) and \(x + 7y = -10\).

Step 3: Set up a new system of equations

The problem asks for points closest to the origin (0,0). The distance from (0,0) to any point \((x, y)\) on a line \(x + 7y = c\) is given by the formula \(\frac{|c|}{\sqrt{1^2 + 7^2}} = \frac{|c|}{\sqrt{50}}\). Set \(c = 10\) and \(c = -10\) for each line separately to minimize the distance.

Step 4: Calculate distance for each line

For the line \(x + 7y = 10\), the distance to the origin is \(\frac{10}{\sqrt{50}}\). Similarly, for \(x + 7y = -10\), the distance to the origin is also \(\frac{10}{\sqrt{50}}\). Since the distances are equal, calculate the intersection points of the line with the circle.

Step 5: Find intersection points

Solve the system of equations for each case: **For line \(x + 7y = 10\):**1. Substitute \(x = 10 - 7y\) into the equation \((x + 7y)^2 = 100\). 2. Simplify to find the possible \((x, y)\) pairs. **For line \(x + 7y = -10\):**1. Substitute \(x = -10 - 7y\) into the equation \((x + 7y)^2 = 100\). 2. Simplify to find possible \((x, y)\) pairs.

Step 6: Calculate possible intersections

For line \(x + 7y = 10\):1. Substitute \(x = 10 - 7y\) into the transformed equation yielding \((x + 7y) = 10\). Resulting intersection: \((10, 0)\).For line \(x + 7y = -10\):1. Substitute \(x = -10 - 7y\) into the transformed equation yielding \((x + 7y) = -10\). Resulting intersection: \((-10, 0)\).

Step 7: Select the points of minimum distance

Calculate the Euclidean distance of each intersection point from the origin. - Distance of \((10, 0)\) from the origin: \(10\).- Distance of \((-10, 0)\) from the origin: \(10\).Both points are equidistant from the origin, so they both are the points of minimum distance.

Key Concepts

Conic Sections

Conic sections are the curves you get when you slice a double cone with a flat plane. In our exercise, we encounter the equation \(x^2 + 14xy + 49y^2 = 100\), which is a type of conic section.
There are different kinds of conic sections: circles, ellipses, parabolas, and hyperbolas.

• A circle is formed when the cutting plane is parallel to the base of the cone.
• An ellipse occurs when the plane cuts through the cone at an angle, not parallel to the base.
• A parabola is shaped when the plane is parallel to the slant of the cone.
• A hyperbola appears when the plane cuts through both halves of the double cone.

Understanding these shapes helps in analyzing and solving problems related to their equations. In our case, rewriting \((x + 7y)^2 = 100\) helped identify the equation as representing lines, specifically two parallel lines \(x + 7y = 10\) and \(x + 7y = -10\). It simplifies the pursuit to find points closest to the origin.

Distance Formula

To find how far one point is from another, the distance formula is a handy tool. It guides us in determining the shortest path between two points in a plane.
The formula is derived from the Pythagorean theorem, expressed as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]For our exercise, we're specifically measuring how far points on the described lines are from the origin \, \((0,0)\).
Since the formula helps measure direct distance in flat geometry, it offers:\[ d = \frac{|c|}{\sqrt{a^2 + b^2}} \] for a line equation of the form \(ax + by = c\).

• Here, \(a = 1\), \(b = 7\), so \(\sqrt{1^2 + 7^2} = \sqrt{50}\).
• Calculating gives equal distances of \(\frac{10}{\sqrt{50}}\) from both lines.

This ensures both lines are equally distanced from the origin, leading us to find precise intersection points with minimum distance on each line.

System of Equations

A system of equations consists of multiple equations that are solved together, all at once. This concept is crucial here, where we deal with lines intersecting at certain points on our curve.
In solving our exercise, each of the transformed lines \(x + 7y = 10\) and \(x + 7y = -10\) forms part of a different system with the equation \((x + 7y)^2 = 100\). To find the closest points:

• Substitute: Solve for \(x\) in terms of \(y\) based on the lines.
• Insert: Plug back into the quadratic form \((x + 7y)^2 = 100\) to maintain consistency of the system.
• Simplify: Rearrange and solve for intersection coordinates \((x, y)\).

The solutions, \, \((10, 0)\) and \, \((-10, 0)\), reveal points on each line nearest the origin—showing that strategic setups of systems can help achieve clarity in problem-solving.