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