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Chapter 5: Problem 6

Find the point on the line \(y=2 x+1\) that is closest to the point (5,2)

Short Answer

Expert verified
The point on the line \(y = 2x + 1\) that is closest to the point (5, 2) is \(\left(\frac{7}{5}, \frac{19}{5}\right)\).

Step by step solution

01

Write the equation of the line in a form that includes both x and y

The given equation of the line is \(y = 2x + 1\). We can think of any point on this line as \((x, 2x + 1)\).
02

Use the distance formula

The distance formula to find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Here, we want to find the distance between the point (5, 2) and any point \((x, 2x + 1)\) on the line. So, we have \(x_1 = 5\), \(y_1 = 2\), \(x_2 = x\), and \(y_2 = 2x + 1\). Plugging these values into the distance formula, we get: \[D = \sqrt{(x - 5)^2 + (2x + 1 - 2)^2}\] \[D = \sqrt{(x - 5)^2 + (2x - 1)^2}\]
03

Minimize the distance using calculus

We want to minimize the distance D. Instead of minimizing D, we can minimize the square of the distance, since squaring is a monotonically increasing function. So, let's find the derivative of \(D^2\) with respect to x and set it to 0 to find the minimum distance. \[D^2 = (x - 5)^2 + (2x - 1)^2\] To find the derivative, we use the power rule and the chain rule. \[\frac{d(D^2)}{dx} = 2(x - 5)(1) + 2(2x - 1)(2)\] \[\frac{d(D^2)}{dx} = 2x - 10 + 8x - 4\] Now, set the derivative to 0 and solve for x. \[2x - 10 + 8x - 4 = 0\] \[10x = 14\] \[x = \frac{7}{5}\]
04

Find the corresponding value of y

Now, we've found the x-value of the closest point. We can use the equation of the line to find the corresponding y-value. \[y = 2x + 1\] Plug in the value of x we found in step 3. \[y = 2 \cdot \frac{7}{5} + 1\] \[y = \frac{14}{5} + 1\] \[y = \frac{19}{5}\]
05

Write the final answer as the closest point

Now that we have both x and y values of the closest point on the line, we can write our final answer. The point on the line \(y = 2x + 1\) that is closest to the point (5, 2) is \(\left(\frac{7}{5}, \frac{19}{5}\right)\).

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