Problem 74 - Precalculus

Question

Find the distance between the point and the line. $$\begin{array}{cc}\text{Point} && \text{Line} \\ (-5,-3) && -2 x-6 y=7\end{array}$$

Step-by-Step Solution

Answer

The distance of the point (-5,-3) to the line -2x-6y=7 is given by \(\frac{21*\sqrt{10}}{20}\)

Step 1: Identify the coefficients of the line equation

First, write the line equation in the standard form Ax+By+C=0. So, -2x-6y=7 should be rewritten as -2x-6y-7=0. Now, identify the coefficients A=-2, B=-6 and C=-7

Step 2: Insert the values into the formula

Plug the coordinates of the point(-5,-3) and the values of A, B and C into the distance formula. Hence, the equation becomes: \[D = \frac{|-2*(-5) -6*(-3) -7|}{\sqrt{(-2)^2 + (-6)^2}}\]

Step 3: Solve the equation

Perform the calculations in both numerator and denominator. This gives: \[D = \frac{|10+18-7|}{\sqrt{4+36}} = \frac{|21|}{\sqrt{40}} = \frac{21}{\sqrt{40}}\]

Step 4: Simplify

To simplify further, notice that \sqrt{40} is equivalent to 2*\sqrt{10}. So, the distance D can be simplified as \[D = \frac{21}{2*\sqrt{10}} = \frac{21}{2}*\frac{1}{\sqrt{10}} = \frac{21}{2}*\frac{\sqrt{10}}{10} =\frac{21*\sqrt{10}}{20}\]

Key Concepts

Line Equation

When dealing with lines in coordinate geometry, understanding the line equation is crucial. In general, a line equation can be expressed in the form of Ax + By + C = 0, where A, B, and C are constants. This is known as the standard form of a line equation.
To convert any line equation to this form, aim to have all terms on one side of the equation, while zero is on the other. This allows for easier identification of coefficients, which are essential in various calculations, such as finding distances.
For example, the line equation -2x - 6y = 7 can be rewritten as -2x - 6y - 7 = 0. Here, the coefficients are clear: A = -2, B = -6, and C = -7. Recognizing these constants helps to utilize formulas in coordinate geometry effectively.

Distance Formula

The distance formula is a tool that helps to find how far apart a point is from a line. It is given by: \[ D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]Where:

• A, B, and C are the coefficients from the line equation Ax + By + C = 0.
• (x₁, y₁) are the coordinates of the given point.

This formula calculates the shortest (perpendicular) distance from a point to a line. It involves plugging the values of the point's coordinates and the line's coefficients into the formula.
For instance, when finding the distance from the point (-5, -3) to the line given by the equation -2x - 6y = 7, we substitute the known values into the formula.
The calculations within both the numerator and denominator help to yield the perpendicular distance, which is the simplest form of distance in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics where geometric problems are solved using algebraic equations. These equations involve coordinates on a Cartesian plane, which makes it easier to handle various geometrical shapes and lines.
The beauty of coordinate geometry is its ability to convert complex geometric relationships into manageable algebraic expressions. By using well-defined formulas, such as the line equation and distance formula, this field aids in understanding space in a numerical context.
Coordinate geometry was employed in the exercise to find the distance between a point and a line. By identifying key elements, such as line coefficients and point coordinates, algebraic calculations were performed efficiently. This approach provides a streamlined way to deal with geometrical issues while reinforcing mathematical concepts.