Problem 51 - Precalculus with Limits

Question

In Exercises 51-58, find the distance between the point and the line. \(\textit{Point}\) \((0, 0)\) \(\textit{Line}\) \(4x + 3y = 0\)

Step-by-Step Solution

Answer

The distance between the given point and the line is 0.

Step 1: Identify Values

Identify the values from the question: \((x_1,y_1) = (0,0)\) and the line equation \(4x + 3y = 0\) gives \(A = 4\), \(B = 3\) and \(C = 0\).

Step 2: Insert Into Formula

Insert the values identified in step 1 into the distance formula: \(d = \frac{|4*0 + 3*0 + 0|}{\sqrt{4^2 + 3^2}} = \frac{0}{\sqrt{16 + 9}} = 0\).

Step 3: Compute Final Answer

Simplify the previous representation to get final answer. The distance is 0 because the point is on the line. This is because when we apply the coordinates of the point to the line equation we get a valid statement (0=0 in this case).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures through the use of algebra and the Cartesian coordinate system. In its essence, coordinate geometry links algebraic equations to geometric curves and shapes by representing them with coordinates on a plane.

In the context of the problem presented, coordinate geometry allows us to translate the problem of finding the distance from a point to a line into an algebraic challenge. We utilize the Cartesian coordinate plane to plot the point \(0, 0\) and to visualize the line \(4x + 3y = 0\). Through the coordinate system, we can use algebraic methods to calculate distances and relations between these geometric entities.

• The \(x\)-axis and \(y\)-axis form the basis of the coordinate system and are perpendicular to each other.
• Every point in the plane is defined by an ordered pair of numbers, representing its coordinates.
• Lines in the plane can be represented by linear equations, showing the relationship between \(x\) and \(y\) coordinates of the points on the line.

Distance Formula

The distance formula is an application of the Pythagorean theorem in coordinate geometry that calculates the straight-line distance between two points in a plane. When it comes to finding the distance from a point to a line, rather than to another point, we modify the formula accordingly.

The distance \(d\) from a point \( (x_1, y_1) \) to a line \(Ax + By + C = 0\) is given by the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \
\]By applying absolute value to the numerator, we ensure that distance is always a non-negative value, as it represents the magnitude of separation.

• This formula derives from the general equation of a straight line and the concept of perpendicular distance from a point to a line.
• The absolute value in the formula reflects that distance is considered without direction.
• The denominator \(\sqrt{A^2 + B^2}\) is the length of the vector perpendicular to the line, which helps in computing the shortest distance—the perpendicular distance.

Using this formula for our exercise provides a methodical approach to quantifying the distance, which in the case of the point \(0, 0\) to the line \(4x + 3y = 0\) is zero, indicating the point lies exactly on the line.

Algebraic Representation

Algebraic representation is a way to describe mathematical concepts and objects using symbols and numbers. It is a language that allows us to succinctly express and manipulate mathematical ideas to solve problems. In algebraic representation, equations and expressions condense complicated relationships into manageable forms.

In regard to our given exercise, the algebraic representation makes it clear why the distance from the point \(0, 0\) to the line \(4x + 3y = 0\) is zero. To analyze this, we look at the equation of the line and the coordinates of the point:

• The equation \(4x + 3y = 0\) has constants \(A = 4\), \(B = 3\), and \(C = 0\) which are used in the algebraic distance formula.
• Substituting the point's coordinates \( (x_1, y_1) = (0, 0) \) into the line’s equation yields a true statement, indicating the point satisfies the line's equation.
• The algebraic procedure of plugging into the distance formula simplifies to a zero distance when the point lies on the line.

Overall, algebraic representation aids in translating geometric problems into algebraic ones, facilitating their solution through established algebraic methods such as the distance formula.