4x + 7y - 5 = 0
7y = -4x + 5
y = (-4/7)x + 5/7
Distance to origin = sqrt( x^2 + y^2 )
D = sqrt( x^2 + ( (-4/7)x + 5/7 )^2 )
D = sqrt( x^2 + (16/49)x^2 + 25/49 - (40/49)x )
D = sqrt( (65/49)x^2 - (40/49)x + 25/49 )
I don't know what level of math you're in, so I don't know if you should know about the following or not.
There's a thing called a "derivative" that tells you the rate of change (the slope) of an equation. If you want to find a minimum or maximum for a function, you can figure out the derivative, set that = to 0, then figure out the x value that makes the derivative = 0.
With this problem we can look at the graph (here: https://www.google.com/?gws_rd=ssl#q=plot(+sqrt(+x%5E2+%2B+(+(-4%2F7)x+%2B+5%2F7+)%5E2+)+) ) and see that we should get a minimum around x = 0.3.
The derivative D ' of D:
D ' = (1/2)( (130/49)x - 40/49 )/sqrt( (65/49)x^2 - (40/49)x + 25/49 )
We want D ' = 0
The (1/2) can't make it 0 and the sqrt stuff on the bottom can't make it 0 (can't make 0 from the bottom of a fraction), so the only part of D ' that can make the whole thing 0 is:
(130/49)x - 40/49 = 0
130x - 40 = 0
130x = 40
x = 4/13 (about 0.3077)
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But there might be a problem if x = 4/13 makes the bottom = 0, so we need to check that.
Does sqrt( (65/49)x^2 - (40/49)x + 25/49 ) = 0 if x = 4/13 ?
sqrt( (65/49)x^2 - (40/49)x + 25/49 ) = 0
(65/49)x^2 - (40/49)x + 25/49 = 0
(65/49)(4/13)^2 - (40/49)(4/13) + 25/49 = 0
(65*16)/(49*169) - 160/(49*13) + 25/49 = 0
1040/8281 - 160/637 + 25/49 = 0
1040/8281 - 2080/8281 + 4225/8281 = 0
1040 - 2080 + 4225 = 0
3185 = 0
Not true, so x = 4/13 does not make the bottom of D ' = 0.
.
So x = 4/13 makes D ' = 0 (a "local minimum" and a "global minimum"). Now we need to know what value for y goes with x = 4/13.
y = (-4/7)x + 5/7
y = (-4/7)(4/13) + (5/7)(13/13)
y = -16/91 + 65/91
y = 49/91
y = 7/13
Now we know x = 4/13 and y = 7/13. This should give us the shortest distance from the line L to the origin (0,0).
Distance to origin = sqrt( x^2 + y^2 )
D = sqrt( (4/13)^2 + (7/13)^2 )
D = sqrt( 16/169 + 49/169 )
D = sqrt( 65/169 )
D = sqrt( 5/13 )
D = about 0.6202