Chapter 3: Q. 85 (page 157)

Find the point on the line that is closest to the point $(3, 1) .$
[Hint: Express the distance $d$ from the point to the line as a function of $x,$ and then find the minimum value of $[d {(x)}^{2}] .$

Short Answer

Expert verified

The point the line closest to $(3, 1)$ is $(2, 2) .$

Step by step solution

Step 1. Given information

We need to determine the point on the line $y = x$ that is closest to the point $(3, 1) .$ 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

We are searching for the line $y = m x + c$ perpendicular to the line $y = x .$ The orthogonal lines have an inversely proportional slope.

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

$m = - 1 .$

So, the line having the slope $1$ and passing through the point $(3, 1) .$

$(y - 1) = - 1 (x - 3) .$

$y = - x + 4 .$

Now to get the coordinate of the point $(x, x)$ is the line $y = x,$ solve.

$x = - x + 4 .$

$2 x = 4 .$

$x = 2 .$

The point on the line closest to the $(3, 1)$ is $(2, 2) .$

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Find the point on the line that is closest to the point $(3, 1) .$
[Hint: Express the distance $d$ from the point to the line as a function of $x,$ and then find the minimum value of $[d {(x)}^{2}] .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Simplify

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Find the point on the line that is closest to the point 3,1.[Hint: Express the distance dfrom the point to the line as a function of x,and then find the minimum value of dx2.

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

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Find the point on the line that is closest to the point $(3, 1) .$
[Hint: Express the distance $d$ from the point to the line as a function of $x,$ and then find the minimum value of $[d {(x)}^{2}] .$