Problem 61 - Analytical Geometry: 2D and 3D

Question

Are the points \((3,4)\) and \((2,-6)\) on the same or opposite sides of the line \(3 x-4 y=8 ?\)

Step-by-Step Solution

Answer

Answer: The points (3, 4) and (2, -6) are on opposite sides of the line 3x - 4y = 8.

Step 1: Substitute the coordinates of the first point

Start by plugging in the coordinates of \((3, 4)\). \(3(3) - 4(4)=8\)

Step 2: Check if the equation is true for the first point

Calculate the left side of the equation for the first point. \(9 - 16 = -7\) The equation is not true for the first point, meaning \((3, 4)\) is not on the line.

Step 3: Substitute the coordinates of the second point

Now, proceed by plugging in the coordinates of \((2, -6)\). \(3(2) - 4(-6)=8\)

Step 4: Check if the equation is true for the second point

Calculate the left side of the equation for the second point. \(6+24=30\) The equation is not true for the second point either, meaning \((2, -6)\) is also not on the line.

Step 5: Determine if the points are on the same or opposite sides of the line

Since both points are not on the line, it means they are either on the same side or on opposite sides. This can be determined by examining their values when plugged into the equation: - For \((3,4)\), the left side of the equation yielded \(-7\) and the right side of the equation is \(8\). - For \((2,-6)\), the left side of the equation yielded \(30\) and the right side of the equation is \(8\). Since one point yielded a result that is less than the constant term (\(8\)) and the other yielded a result that is greater than the constant term, these points are on opposite sides of the line.

Key Concepts

Points and Lines

In coordinate geometry, points are defined using coordinates, represented as pairs \(x, y\).
A line, on the other hand, can be represented by a linear equation in the form \(Ax + By = C\).
This equation acts like a boundary in the plane. By plugging a point's coordinates into the line's equation, we can determine its relation to the line.Here's how it works:

• If you plug a point's coordinates into the line equation and the result equals the constant term, the point is on the line.
• If the result is less than the constant on the right side, the point is on one side of the line.
• If the result is greater, it's on the opposite side.

Understanding points and lines is crucial in solving geometric problems involving location and distance.

Equation of a Line

An equation of a line, such as \(3x - 4y = 8\), represents all the points that lie exactly on the line.
For any specific line in a plane, this equation highlights the relationship between the x and y coordinates of a point.
To understand this concept better, let's take the line \(3x - 4y = 8\):

• This equation consists of coefficients \(3\) for \(x\) and \(-4\) for \(y\), illustrating the slope relationship between these two variables.
• The constant \(8\), represents where the line intercepts or crosses the other axes.

Checking if a point lies on a line involves substituting the point's coordinates into this equation. If the resulting value on the left matches the constant term on the right, the point is on the line. Otherwise, it is not.

Inequalities in Geometry

In coordinate geometry, inequalities help in understanding spatial relationships, especially when determining which side of a line a point resides.
A line divides the plane into two regions, with each region represented by an inequality.Here’s what happens with the inequalities:

• If substituting a point in an equation results in a value less than the equation's constant, the point lies in the region described by one inequality (\(3x - 4y < 8\)).
• If it results in a value greater, it’s in the opposite region described by another inequality (\(3x - 4y > 8\)).

This concept helps us not just identify the position, but also solve problems related to regions designated by lines in geometry. Therefore, by examining and comparing these inequalities, we can conclude which side of the line different points are located.