Can You Find the Point P on Line y3x That is Closest to the Point (30, 3)?

To find the point P on the line y 3x that is closest to the point (30, 3), we can use the method of minimizing the distance between the point P and (30, 3).

Step 1: Define the Point P

Let the coordinates of point P be (x, y). Since P lies on the line y 3x, we can express P as P (x, 3x).

Step 2: Distance Formula

The distance D between the points P(x, 3x) and (30, 3) can be expressed using the distance formula:

D sqrt{(x - 30)^2 (3x - 3)^2}

Step 3: Simplify the Distance

We can simplify the expression inside the square root:

D sqrt{(x - 30)^2 (3x - 3)^2}

sqrt{x^2 - 6 900 9x^2 - 18x 9}

sqrt{1^2 - 78x 909}

Step 4: Minimize the Distance

Instead of minimizing D, we can minimize D^2 to avoid dealing with the square root:

D^2 (x - 30)^2 (3x - 3)^2

x^2 - 6 900 9x^2 - 18x 9

1^2 - 78x 909

Step 5: Find the Minimum Value

To find the minimum, we can take the derivative of D^2 and set it to zero:

frac{dD^2}{dx} 2 - 78 0

x frac{78}{20} 3.9

Step 6: Find the y-coordinate for P

Now substitute x back into the equation of the line to find y:

y 3x 3 * 3.9 11.7

Thus, the point P is (3.9, 11.7).

Step 7: Calculate the Least Distance between P and (30, 0)

Now we need to find the distance between P(3.9, 11.7) and (30, 0):

D sqrt{(3.9 - 30)^2 (11.7 - 0)^2}

sqrt{(-26.1)^2 11.7^2}

sqrt{681.21 136.89}

sqrt{818.1}

approx 28.6

Conclusion

The point P on the line y 3x that is closest to (30, 3) is (3.9, 11.7) and the least distance between P and (30, 0) is approximately 28.6 units.

This problem also shows another method by finding the intersection of the line y3x and the perpendicular line from point (30, 3). The perpendicular line passes through (30, 3) with the equation 3y -x 39. Solving these two equations gives the same point P (3.9, 11.7).

Keywords: optimization, distance formula, line closest point