# Find the point on the line y = 2x + 4 that is closest to the origin.?

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• c = √((x-0)² + (y-0)²)

c = √(x² + y²)

c = √(x² + (2x+4)²)

c = √(x² + 4x² + 16x + 16) = √(5x² + 16x + 16)

dc/dx = (10x + 16) / 2√(5x²+16x+16) = 0

10x + 16 = 0

x = -16/10 = -8/5

y = 2(-8/5) + 4 = (-16/5) + 4 = 4/5

(-8/5, 4/5) is the point closest to the origin

• y = 2x + 4

m = 2

Perpendicular line:

m = –(1/2)

Equation of perpendicular line parring through origin.

(y – 0) = –(1/2)(x – 0)

y = –(1/2)x

2y = –x

x + 2y = 0

Find intersection with given line:

x + 2(2x + 4) = 0

x + 4x + 8 = 0

5x = –8

x = –8/5

y = 2(–8/5) + 4 = –16/5 + 4 = 4/5

(x, y) = (–8/5, 4/5) …. point on y = 2x + 4 closest to origin.

• Easy, simple differentiation problem:

s := distance at (x, y)

s = sqrt(x^2+y^2) = sqrt[x^2+(2x+4)^2]

s = sqrt[5x^2+16x+16]

ds/dx = (1/2)(10x+16)/sqrt[5x^2+16x+16]

set ds/dx == 0 solve for x

10x+16 == 0

x = -8/5

y(-8/5) = 4/5

To show that this is a min and not a max calculate d^2 s/dx^2 and evaluate it at x=-8/5. If that value is positive then that point is a minimum.

• The closest point is the point which is on the line which makes 90 degree with the line which passes through (0,0)

Let it (a,b)

Now

b=2a+4 ——(1)

And now find the second equation using fact that the line which passes (0,0) & (a,b) makes 90 degree with the line

y = 2x + 4

Now u have 2 equations n 2 unknowns. Finding a, b will b the solution

• step 1: find the x and y intercept. if it crosses the origin then you are done.

Otherwise to find the point closest to the origin we need the x and y intecepts

So

x-int: 0=2x+4 implies xint is -2

y-int: y=2(0)+4 implies yint is 4

so the point closest to the origin is half the distance from -2 to 0 in the x coordinates and half the distance from 0 to 4 in the y coordinates

i.e. (-1,2) is the point closest to the origin

• take the point as (h,k)

so k = 2h + 4————–(1)

using distance formula , distance of (h,k) from origin is

sqrt (h² + k²)

let z = distance²

z = h² + k² = h² + (2h + 4)² using equation (1)

now find the min value of z using calculus or any other method

you could refer the following link for some information on calculus method

• the line crossing this line that is perpendicular to it and passes through the origin is what you want

the product of the slopes of a line and one that is perpendicular to it is -1, so the slope of the line the intersects this one is

(y-0) = -0.5 (x – 0) (the slope is – 1/2 )

solve this like WAHEED indicates

• y = 2x + 4

slope of line perpendicular to it = -0.5, equation of it passing through origin.

y = -0.5 x

intersect with original line in point:

– 0.5x = 2x + 4

-2.5 x = 4

x = – 1.6, y = -3.2 + 4 = – 0.8

the required point is (- 1.6, – 0.8) ans

• graph it and find the point closest to the origin. how hard could it be?

• there is a formula which you can get that distance easily

i don’t remember the expression but you could get it in an analytic geometry book

and also you can get that measure with a graphical method

try it and you could make your own formula

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