Need help with distance problem

[Timcago]

Math course : college algebra

Problem:

What point on the graph of y=3x+4 is closest to the point (1,2)?

Here are the hints it gives me

(1) you will need to use the distance forumula
(2) y=(the square root of)f(x) is minimized when y=f(x) is minimized

Does anyone know how to do this? We are learning about minimum and maximum when it comes to functions.

 

[stapel]

Use the square of the distance formula:

. . . . .d² = (x₁ - x₂)² + (y₁ - y₂)²

Note that your points are (1, 2) and (x, 3x + 4).

Eliz.

 

[Timcago]

I get

The square root of ((5(2x^2+2x+1)))

How does that help me get the point on the line closest to (1,2)?

 

[Timcago]

Anyone know how to solve this one?

 

[pka]

The line \(\displaystyle y - 2 = \frac{{ - (x - 1)}}{3}\) contains the given point and is perpendicular to the given line.
Find the point of intersection of these two lines.
That is the closest point.

 

[skeeter]

What point on the graph of y=3x+4 is closest to the point (1,2)?

actually, you don't need the distance formula to solve this.

The point on the line y=3x+4 closest to (1,2) will lie on the line perpendicular to y=3x+4 and passing through (1,2).

since y=3x+4 has slope m=3, a line perpendicular to y=3x+4 will have slope m=-1/3.

using the point-slope form of a linear equation with m=-1/3 and the point (1,2) ...

y - 2 = (-1/3)(x - 1)
y - 2 = (-1/3)x + (1/3)
y = (-1/3)x + (7/3)

the point you're looking for is the intersection of the two lines ... so set them equal and solve for x ...

3x + 4 = (-1/3)x + (7/3)
9x + 12 = -x + 7
10x = -5
x = -1/2

since y = 3x+4 ...

y = 3(-1/2) + 4 = 5/2

the point on the line y = 3x+4 closest to the point (1,2) is (-1/2, 5/2).

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now, if you insist upon the distance formula method ...

D = sqrt[5(2x² + 2x + 1)]

if (2x² + 2x + 1) is minimized, then D will be minimized.

(2x² + 2x + 1) is a parabola that opens upward ... it's vertex (or minimum value in this case) is located at x = -b/(2a) = -2/4 = -1/2

x = -1/2 is where the distance is minimized ... find the y-value on the line y=3x+4 by substituting -1/2 in for x and evaluating y.