Need help please!! Find the point on the line y=5x+4 that is closest to the origin?
Author : david
Submitted : 20180115 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
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 √( ( x0)² + (y0)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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