Find the point on the line y = 5x + 4 that is closest to the origin.?

Question

Find the point on the line y = 5x + 4 that is closest to the origin. (Give your answers correct to three decimal places.)

Answer 1

To determine the point on a line that is closest to a given point, look for a point where the line connecting the point and the original line is perpendicular to the original line. So, if (h, k) is the point of closest approach to the origin, then the line connecting (h,k) and the origin should be perpendicular to the line y=5x+4. Two perpendicular lines have slopes that are negative reciprocals, that is their product is -1.



First establish the relationship on the line through the origin.

m=(k-0) / (h-0) = -1/5

k/h = -1/5

Cross-multiply

5k = -h

h = -5k



Now, use that relationship in the original line definition to determine the point of intersection.

y = 5x + 4 at the point of intersection (h, k)

k = 5(h) + 4

k = 5(-5k) + 4

k = -25k + 4

26k = 4

k = 4/26 = 2/13

h= -5k = -5 (2/13) = -10/13



So the point of intersection is (-10/13, 2/13) or to three decimal points, (-0.769, 0.154)

Answer 2

its easy, to understand the theory first draw x axis and y axis and then just draw a line without considering to anything (taking a fair line / don`t ever try to draw the exact 5x+4). Name it as y = 5x+4.

Now, draw a perpendicular to y=5x+4 line, which connects the origin('O') and this y=5x+4 you drwn.

Name the point that these two line meet as 'T".

So now you see that T is the point you want.

if T=(x',y')

THEN , the line OT is perpendicular to y=5x+4 line. so,

(y'-0)/(x'-0) * 5 = -1



** { this is a proved statement If m1*m2 = -1 ,then any "y=m1x+c1" and "y = m2x+c2" lines are perpendicular.}**



so you get, y' = -x'/5 -----------(1)



the point T is on the line y=5x+4. So,

y' = 5x' +4 ------------(2)



now you can solve (1) & (2) and get the closest point to the origin..



you can use this format to find the closest point to a line from any point. not only origin. Its very easy if its origin. this format again helps in many situations.. so be sure to get the idea not the answer to the question..It will help very much if you are a student..

Answer 3

First, we need to find the line that is perpendicular to y = 5x + 4 that crosses the origin.

The line has the slope of y = - 1/5. And since it crosses the origin, the y intercept of the new line is 0.

The line we're looking for is y = -(1/5)x

Next, solve for x and y using the equations

y = 5x + 4

and

y = -(1/5)x



5x + 4 = -(1/5)x

=> (26/5)x = -4

=> x = -(10/13) = -0.769

Plugging x = -(10/13) back into the original equation, y = 5x + 4, we get

y = 5(-(10/13)) + 4

y = 0.154



So the point on the line y = 5x + 4 that is closest to the origin is the point

P(-0.769, 0.154)



Hope that helped!

Answer 4

The equation of line perpendicular to the given line has a slope = -1/5 and as it will pass through the origin , its equation is y = (-1/5)x. Solving this to get the point of intersection of this line with the given line y = 5x + 4, we get the point of intersection as (-10/13, 2/13). This is the point closest to the origin on the given line.

Answer 5

I can't really say why, but a tangent line passing through (0,0) will be the shortest distance, and the point of intersection will be the point you are looking for.



y = 5x + 4



tangent



y = -1/5x + b



find b



y = -1/5x + b

0 = -1/5(0) + b

0 = b



so the equation is



y = -1/5x



find the intersection



y = -1/5x



y = 5x + 4

-1/5x = 5x + 4

0 = 5x + 1/5x + 4

-4 = 5x + 1/5x

-4 = 5_1/5x

-4 = 26/5x

-4*5/26 = x

-20/26 = x

-10/13 = x

thats approximately -.769



y = 5x + 4

y = 5(-10/13) + 4

y = -50/13 + 4

y = -50/13 + 52/13

y = 2/13

thats approximately .154



ANSWER

(-10/13 , 2/13) or (-.769 , .154)

Answer 6

y = 5x + 4 closest to (0,0)



D = √[(x-0)^2 + (y-0)^2]

plug in what you know for y...

D = √[x^2 + (5x +4)^2]

simplify...

D = √[x^2 + 25x^2 + 40x + 16]

D = √[26x^2 + 40x + 16]

take the derivative...

dD/dx = ((52x +40)/2√[26x^2 + 40x +16])

set the derivative = to 0

0 = ((52x +40)/2√[26x^2 + 40x +16])



(-40/52) is your answer... for x... then you have to find y by plugging this number in for x in the equation y = 5x + 4

Answer 7

(-0.35, 2.25)

I graphed it by using y=mx+b