Question

Find the distance from point A(−1, 7) to the line y= 3x. Round your answer to the nearest tenth.

Answer 1

the line joining A to y = 3x will be perpendicular to the latter so its slope will be -1/3.

its equation will be y - 7  = -1/3(x  + 1)

= y = -1/3x + 20/3

where the lines intersect:-

3x = -1/3x + 20/3
10/3 x  = 20/3

x =  2   and y =   3*2 = 6

so we need to find distance between (-1,7) and (2,6)  


this is  sqrt[((7-6)^2 + (-1-2)^2)]  =  sqrt 10   =  3.2 to nearest tenth.

Answer 2

I assume that we want to find the minimum distance between the point A(-1, 7) and the line y = 3x.

We will get that the distance is 3.22

So first, let's see how to get the distance between two points (x₁, y₁) and (x₂, y₂). We just use the general formula:

$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$

Now we want to find the distance between (-1, 7) and a point on the line y = 3*x, that can be written as (x, 3x)

That distance is:

$d = \sqrt{(-1 - x)^2 + (7 - 3x)^2}$

Now we want to minimize this, which is equivalent to minimize d^2, then we can define:

D = d^2

and write:

$D = (-1 - x)^2 + (7 - 3x)^2\\\\D = 1 - 2x + x^2 + 49 - 42x + 9x^2\\\\D = 10x^2 - 44x + 50$

Now we need to minimize this, which is a parabola, so the minimum will be at the vertex of the parabola (we know this because the leading coefficient is positive, thus the parabola opens upwards).

Remember that for a parabola like:

a*x^2 + b*x + c

The vertex is at:

x = -b/2a

Then for our parabola, the vertex will be at:

x = -(-44)/(2*10) = 44/20 = 2.2

Thus the minimum distance is given by:

$d = \sqrt{(-1 - 2.2)^2 + (7 - 3*2.2)^2} = 3.22$

If you want to learn more, you can read:

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