Problem 71 - Precalculus

Question

Find the distance between the point and the line. $$\begin{array}{cc}\text{Point} && \text{Line} \\ (-1,2) && 5 x+3 y=-4\end{array}$$

Step-by-Step Solution

Answer

The distance from the point \((-1,2)\) to the line \(5x + 3y + 4 = 0\) is \( \frac{1}{\sqrt{34}}\).

Step 1: Identify the given line and point

The given point is \((-1,2)\) and the given line equation is \(5x + 3y = -4\).

Step 2: Write the equation of the line in the form Ax + By + C = 0

The equation \(5x + 3y + 4 = 0\) is already in this form. Now we can identify \(A = 5\), \(B = 3\), and \(C = 4\).

Step 3: Substitute the given values into the distance formula

To find the distance \(d\), substitute the appropriate values into the formula \(d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\) to get \(d = \frac{|5*(-1) + 3*2 + 4|}{\sqrt{5^2 + 3^2}}\).

Step 4: Simplify the equation

Carry out the calculations to find \(d = \frac{1}{\sqrt{34}}\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that enables us to study geometry using a coordinate system. It is a bridge between algebra and geometry, providing us tools and methods to solve problems involving figures, shapes, and lines on a two-dimensional plane. With Coordinate Geometry, it's possible to determine distances, angles, and positions using algebraic equations.

• Points: These are expressed as \(x, y\) pairs representing positions in the plane.
• Lines: Defined by equations, such as \(Ax + By + C = 0\).

Understanding these basic components helps us perform a variety of tasks, including calculating distances and angles between different geometric entities.

Distance Formula

The Distance Formula is a powerful tool in Coordinate Geometry used to determine the distance between two points or a point and a line. For a point \(x_1, y_1\) and a line \(Ax + By + C = 0\), the distance formula is: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}.\]

This formula helps us calculate the perpendicular distance from the point to the line using the given coefficients of the line equation and the coordinates of the point. Its derivation stems from the Pythagorean Theorem, ensuring geometric accuracy.

Line Equation

A Line Equation in the form \(Ax + By + C = 0\) is a linear representation of a straight line in the coordinate plane. In this equation:

• \(A\), \(B\), and \(C\) are constants that define the line's characteristics.
• The equation can be rearranged into more familiar forms like the slope-intercept form \(y = mx + b\).

To find distances or intersections, we often need the equation of the line in the standard form. In the exercise, the given line is \(5x + 3y + 4 = 0\). Recognizing this form is essential for using the Distance Formula effectively.

Perpendicular Distance

The Perpendicular Distance is the shortest distance between a point and a line. It is always at a right angle to the line and is a crucial measure in geometry and physics. Calculating this distance involves:

• Identifying the line's equation and the point on the coordinate plane.
• Using the Distance Formula to compute the shortest path.

In our example, the point \(-1,2\)\ and the line \(5x + 3y = -4\) are used to find this perpendicular distance. By substituting the values into the formula, we obtained \(d = \frac{1}{\sqrt{34}}\), representing the shortest possible distance from the point to the line.