Mathematics: Find the point p on the line yequals6x that is closest to the point left parenthesis 74 comma 0 right parenthesis. what is the least distance between p and left parenthesis 74 comma 0 right parenthesis

Step-by-step explanation:Simplify like terms.10a-2a+3b+b = 8a+4b...

Find the point p on the line yequals6x that is closest

7obadiah

15.03.2022

The least distance would be 73 units ( approx )

Step-by-step explanation:

Let the coordinates of P are (x_1, y_1) ,

Since, point P lies on the line y = 6x

So, P will satisfy the above line,

y_1=6x_1------(1) ,

Now, By the distance formula,

The distance between P and (74,0),

$D=\sqrt{(74-x_1)^2+(0-y_1)^2}$

$=\sqrt{(74-x_1)^2+(-6x_1)^2$ ( From equation (1) )

$=\sqrt{(74-x_1)^2 + 36x_1^2}$

Differentiating with respect to x_1 ,

$D'(x_1)=-\frac{1}{(74-x_1)^2 + 36x_1^2}[-2(74-x_1)+72x_1]$

Again differentiating with respect to x,

$D''(x_1)=\frac{394272}{(36x^2+(74-x)^2)^\frac{3}{2}}$

For maxima or minima,

D'(x_1)=0 ,

$-\frac{1}{(74-x_1)^2 + 36x_1^2}[-2(74-x_1)+72x_1]=0$

-2(74-x_1)+72x_1=0

-148+2x_1+72x_1=0

74x_1=148

x_1=2

At x_1=2 D''(x) = positive

Thus, for x_1=2 D is minimum,

And, minimum distance,

$D(2) = \sqrt{(74-2)^2 + 36(2)^2}=72.9931503636\approx 73$

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