Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 85

Chapter 4, Problem 85

The distance between the two points $P of open paren x sub 1 comma y sub 1 close paren$ and $R of open paren x sub 2 comma y sub 2 close paren$ is $d(P, R) = $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$ . Distance formula. Find the closest point on the line $y equals 2 x$ to the point $open paren 1 comma 7 close paren$ . (Hint: Every point on $y equals 2 x$ has the form $open paren x comma 2 x close paren$ , and the closest point has the minimum distance.)

Verified step by step guidance

Identify the general form of any point on the line y = 2x, which is given as (x, 2x).

Write the distance formula between the point (1, 7) and a general point on the line (x, 2x): $d = \sqrt{(x - 1)^2 + (2x - 7)^2}$.

Since the square root function is increasing, minimize the squared distance instead to simplify calculations: $d^2 = (x - 1)^2 + (2x - 7)^2$.

Expand and simplify the expression for $d^2$ to get a quadratic function in terms of $x$.

Find the value of $x$ that minimizes $d^2$ by taking the derivative with respect to $x$, setting it equal to zero, and solving for $x$. Then, substitute this $x$ back into $y = 2x$ to find the corresponding $y$ coordinate of the closest point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distance Formula

The distance formula calculates the distance between two points in the coordinate plane using their coordinates. It is derived from the Pythagorean theorem and is given by d = √((x₁ - x₂)² + (y₁ - y₂)²). This formula helps quantify how far apart two points are.

Equation of a Line

The equation y = 2x represents a line where every point satisfies the relationship between x and y coordinates. Understanding this allows us to express any point on the line as (x, 2x), which is essential for substituting into the distance formula to find the closest point.

Minimizing Distance Using Calculus or Algebra

To find the closest point on a line to a given point, we minimize the distance function with respect to x. This involves expressing distance as a function of x, then finding its minimum value by setting the derivative to zero or using algebraic methods, ensuring the shortest distance is found.

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