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Need help please!! Find the point on the line y=5x+4 that is closest to the origin?

Author : david

Submitted : 2018-01-15 10:12:45    Popularity:     

Tags: point  Find  line  origin  closest  

The perpendicular line through the origin is .. y = (-1/5)x The point where these lines intersect is .. (-1/5)x = 5x +4 .. x = -4/(26/5) = -10/13 .. y

Answers:

The perpendicular line through the origin is
.. y = (-1/5)x
The point where these lines intersect is
.. (-1/5)x = 5x +4
.. x = -4/(26/5) = -10/13
.. y = (-1/5)*(-10/13) = 2/13

The point on the line y = 5x +4 closest to the origin is (-10/13, 2/13).

What class are you taking? There is more than one way to do this.
Algebra method ... realize that the point is on a perpendicular to the goven line .. y = 5x + 4, so the perp. has slope -1/5 and passes thru the origin. The equation of this line is y = -(1/5)x + 0 (because of the origin) ... then just solve for the intersection of the 2 lines.
-(1/5)x = 5x + 4
-(26/5)x = 4
x = -20/26 = -10/13 ... y = (-1/5)x = (-1/5)(-10/13) = +2/13
... so the point is ( (-10/13), (2/13) )

--- calculus ... use derivative of distance formula
points of the line y = 5x + 4 are (x, 5x+4) ... the origin = (0,0)
distance between points
d^2 = (x - 0)^2 + ((5x + 4) - 0)^2 <<<< you could use the sq root for the dist. instead of d^2, but either will work
f(x) = (x - 0)^2 + ((5x + 4) - 0)^2 ... simplify before deriv.
f(x) = x^2 + 25x^2 + 40x + 16 = 26x^2 + 40x + 14 ... now find deriv.
f '(x) = 52x + 40 = 0
x = -40/52 = -10/13 <<< same answer
... then use this to find the y value of the point, which will be the same
... good luck, pick the method that applies to your class...

y=5x+4 and y=-x/5
solve for x , y

The closest point to the origin is (-4/5,0) = (-0.8, 0)...........ANS
(see graph below)

A point of the line can be expressed as (x,y) .
The origin is (0,0).

The distance between these two points is thus √( ( x-0)² + (y-0)² ).
Replacing y by 5x+4 yields √( x² + (5x+4)²).

Expand what is under the √. Note that its a parabola.
What is the minimum of that parabola? (hint: its vertex).

Show your steps here if need be and we can take it from there! Done!



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