Problem 69 - Precalculus

Question

Find the distance between the point and the line. $$\begin{array}{cc}\text{Point} && \text{Line} \\ (6,2) && -3 x+4 y=-5\end{array}$$

Step-by-Step Solution

Answer

The shortest distance from the point (6,2) to the line -3x+4y=-5 is 3 units.

Step 1: Identify the variables

From the equation of the line, we can identify that \(A = -3\), \(B = 4\), and \(C = -5\). The coordinates of the point are \((x_1, y_1) = (6, 2)\).

Step 2: Substitute into the distance formula

Substitute the values of A, B, C, x1, and y1 into the distance formula. We get \(d = \frac{|-3(6) + 4(2) + (-5)|}{\sqrt{(-3)^2 + 4^2}}\)

Step 3: Simplify and calculate distance

Simplify the above equations to get: \(d = \frac{|-18 + 8 - 5|}{\sqrt{9 + 16}} = \frac{15}{\sqrt{25}} = 3\)

Key Concepts

Distance formula

The distance formula is key in computing the shortest distance from a point to a line in coordinate geometry. It’s akin to finding the shortest path as the crow flies. In general, for a given point \((x_1, y_1)\) and a line \(Ax + By + C = 0\), this formula is useful:

• \(d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\)

This formula derives from the geometric properties of a line and a point. It essentially uses the perpendicular line between the point and line, since the shortest distance is always perpendicular.
This formula deals with absolute value operations to ensure distance is non-negative. The denominator comes from the definition of the slope’s magnitude.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry through the use of a coordinate system. This allows you to analyze the geometric properties of figures and solve problems algebraically.

• Points are expressed as coordinates, like \((x, y)\).
• Lines can be expressed with equations like \(Ax + By + C = 0\).

This approach is exceptionally useful in plotting graphs and understanding geometric transformations. It helps convert visual information into mathematical expressions, making problem-solving more straightforward.
In finding distances or other attributes, coordinate geometry leverages algebraic formulas, offering tools that are both powerful and versatile.

Linear equations

Linear equations are statements of equality between algebraic expressions that provide the basic framework for lines in coordinate geometry. A typical form of a linear equation is \(Ax + By + C = 0\).

• Here, \(A\), \(B\), and \(C\) are constants.
• \(x\) and \(y\) are variables representing points on a graph.

The solutions to a linear equation refer to all the points \((x, y)\) that lie on the line described by the equation. The slope-intercept form, \(y = mx + b\), is another representation that gives insight into the line’s slope \(m\) and y-intercept \(b\).
Understanding linear equations is foundational for tackling more complex mathematical concepts, such as systems of equations and calculus.