Chapter 3: Q. 86 (page 157)

Find the point on the line $y = x + 1$ that is closest to the point $(4, 1) .$

Short Answer

Expert verified

The point on the line $y = x + 1$ closest to $(1, 4)$ is $(2, 3) .$

Step by step solution

Step 1. Given information

We need to determine the point on the line $y = x + 1$ that is closest to the point $(4, 1) .$ We need to express the distance $d$ from the point to the line as a function of $x$ and then find the minimum value of $[d {(x)}^{2}] .$

Step 2. Simplify

Let the line be perpendicular to the line $y = x + 1$ and $y = m x + c .$

Since the perpendicular lines have an inversely proportional slope.

So, $m = - \frac{1}{1} .$

$m = - 1 .$

The line passing through the point $(4, 1)$ and having the slope $- 1$ is,

$y - 1 = - 1 (x - 4) .$

$y = - x + 5 .$

Now put the point $(x, y)$ that is $(x, x + 1)$ in $y = - x + 5 .$

$- x + 1 = - x + 5 .$

$2 x = 4 .$

$x = 2 .$

The point on the line $y = x + 1$ is closest to $(1, 4)$ is $(2, 3) .$

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Find the point on the line $y = x + 1$ that is closest to the point $(4, 1) .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Simplify

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Find the point on the line y=x+1that is closest to the point 4,1.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Pure Maths

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Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find the point on the line $y = x + 1$ that is closest to the point $(4, 1) .$