Question

find the point on the line y = -4x that is closest to the point (5, 0). round answers to 2 decimal places. note: $d = \sqrt{(x - x_0)^2 + (y - y_0)^2}$ ( , ) question help: message instructor submit question jump to answer

Explanation:

Step1: Substitute \(y = - 4x\) into the distance formula

Let the point on the line \(y=-4x\) be \((x,y)=(x, - 4x)\), and the given point is \((x_0,y_0)=(5,0)\). Then the distance \(D=\sqrt{(x - 5)^2+(-4x-0)^2}=\sqrt{(x - 5)^2 + 16x^2}\). To simplify the calculation, we can minimize \(D^{2}\) instead (since minimizing \(D^{2}\) is equivalent to minimizing \(D\) for \(D\geq0\)). Let \(f(x)=(x - 5)^2+16x^2\).
Expand \(f(x)\):

$$
LATEXBLOCK0
$$

Step2: Find the derivative of \(f(x)\)

Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \(f^\prime(x)=(17x^{2}-10x + 25)^\prime\).
\(f^\prime(x)=34x-10\)

Step3: Set the derivative equal to zero and solve for \(x\)

Set \(f^\prime(x)=0\), then \(34x-10 = 0\).

$$
LATEXBLOCK1
$$

Step4: Find the \(y\) - coordinate

Since \(y=-4x\), when \(x = \frac{5}{17}\), \(y=-4\times\frac{5}{17}=-\frac{20}{17}\approx - 1.18\)

Answer:

\((0.29,-1.18)\)