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# Find the Point on the Straight Line 2x+3y = 6, Which is Closest to the Origin. - Mathematics

Sum

Find the point on the straight line 2x+3y = 6,  which is closest to the origin.

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#### Solution

The equation of line is given as 2x+3y=6
therefore y= (6 -2x)/3
therefore The point of the line can be taken as P =( x,(6-2x)/3)

Co -ordinate of origin are O = (0,0)
Co-ordinates of origin are (0,0)= O

OP = sqrt((x-0)^2 + ((6-2x)/3 -0))^2

∴ OP = sqrt(x^2 + (2- (2x)/3)^2

∴ OP^2 = x^2 + (4x^2)/9+4-(8x)/3

OP^2 = (13x^2)/9 - (8x)/3 + 4

Let 2 ( ), OP f x OP  is minimum when OP2 is minimum.
therefore f(x) = (13x^2)/9 - (8x)/3 + 4       ....(1)
Differentiating w.r.t. ‘x’ we get,

f'(x)= 26/9>0

therefore OP is  minimum at x = 12/13
therefore y = 2-(2x)/3

therefore y = y - 2/3xx12/13 = 2- 8/13= (26-8)/13

therefore y=18/13
∴ The closest point on the line 2 3 6 x y   with origin is (12/13 18/13)

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