Problem 67 - Precalculus

Question

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

Step-by-Step Solution

Answer

The distance between the point (2,3) and the line 3x+y=1 is \(\frac{4\sqrt{10}}{5}\).

Step 1: Reformat the Line Equation

To apply the formula, we need to write the given line equation in the format Ax+By+C=0. We can rewrite 3x+y=1 to 3x+y-1=0. So, A=3, B=1, C=-1.

Step 2: Plug into Distance Formula

Next, we plug these values of A, B, and C and the coordinates of the point (2,3) into the distance formula \(d = \frac{|Ax+By+C|}{\sqrt{A^2+B^2}}\). It comes out to be \(d = \frac{|3(2)+1(3)-1|}{\sqrt{3^2+1^2}}\).

Step 3: Simplify the Equation

We simplify the equation to find the distance. It leads to \(d = \frac{6+3-1}{\sqrt{9+1}} = \frac{8}{\sqrt{10}}\).

Step 4: Rationalize the Denominator

Then we rationalize the denominator for the under root term. It gives \(d = \frac{8\sqrt{10}}{10} = \frac{4\sqrt{10}}{5}\).

Key Concepts

Distance Formula

The distance formula helps us calculate the shortest path between a point and a line. It is a straightforward tool, derived from the Pythagorean theorem, that allows us to measure this perpendicular distance. We use it when dealing with a point

• (x₁, y₁) with coordinates

and a line given by the equation

• Ax + By + C = 0.

The specific formula for calculating this distance is:\[d = \frac{|Ax₁ + By₁ + C|}{\sqrt{A^2 + B^2}}\]This setup helps in considering both the direction and the scalar nature of the coefficients involved.
To determine the distance using this equation, substitute the coordinates of the point and the coefficients from the line's equation directly into the formula. It's important to notice how the numerator

• |Ax₁ + By₁ + C| represents the absolute perpendicular magnitude

while the denominator

• \(\sqrt{A^2 + B^2}\) ensures it's being scaled properly relative to the line's slope.

This tool is essential in analytic geometry, offering clarity and precision without needing to graph geometrically.

Equation of a Line

An understanding of the equation of a line is pivotal in calculating the distance to a point. Lines in mathematics are represented using the slope-intercept formula:

• y = mx + b, where m is the slope and b is the y-intercept.

However, to use the distance formula, we must reconfigure it to the standard form:

• Ax + By + C = 0.

The transition between these forms is simple - rearrange terms so the equation equals zero. For instance, a line initially expressed as

• 3x + y = 1,

becomes

• 3x + y - 1 = 0,

thus forming the structure necessary to plug values into our distance formula. Here,

• A = 3,
• B = 1,
• C = -1.

Such translations are crucial for simplifying analytical tasks, aligning any given line correctly when calculating distances or intersections.

Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate any irrational numbers from the bottom of a fraction. This technique is particularly helpful when we strive for clarity and standard forms in answers, making results easier to read or use in further calculations.
In the given exercise, the distance was initially calculated as:\[d = \frac{8}{\sqrt{10}}\]However, to rationalize, you multiply both numerator and denominator by the square root present in the denominator. Therefore:\[d = \frac{8\sqrt{10}}{10}\]By doing this, the denominator becomes a rational number. Further simplifications can reduce it even further to

• \(\frac{4\sqrt{10}}{5}\).

Rationalizing is essential to present the simplest form of an answer, adhering to mathematical conventions and avoiding irrational aspects whenever possible. It's a neat trick to polish the final solution in, quite literally, clearer terms.