12 Answers

c = √((x0)² + (y0)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
xint: 0=2x+4 implies xint is 2
yint: 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
(y0) = 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