# Find the point on the line 2x + 3y - 1 = 0 which is closest to the point (-1, -1)?

Relevance

2x + 3y - 1 = 0

3y = -2x + 1

y = (-2/3)x + 1/3

the point on the line closest to (-1 , -1) will lie on the perpendicular line passing through that given point

slope of line is -2/3, so the perpendicular line will have slope 3/2 and contain (-1 , -1)

thus, -1 = 3/2 (-1) + b

-1 = -3/2 + b

1/2 = b

the perpendicular line through (-1 , -1) is y = 3/2 x + 1/2

now you can solve these two simultaneously to find the point of intersection

y = 3/2 x + 1/2 = (-2/3) x + 1/3

(3/2 + 2/3)x = 1/3 - 1/2 = 2/6 - 3/6 = -1/6

(9/6 + 4/6) x = -1/6

13/6 x = -1/6

13x = -1

x = -1/13

if x = -1/13, then y = (3/2)(-1/13) + 1/2 = -3/26 + 1/2 = -3/26 + 13/26 = 10/26 = 5/13

closest point: (-1/13 , 5/13)

edit: distance from (0 , 1/3) to (-1 , -1)

sqrt[1^2 + (4/3)^2] = sqrt[1 + 16/9] = sqrt(25 / 9) = 5/3 = 1.667

intersection of y = 3/2 x + 1/3 and y = -2/3 x + 1/3 is (0 , 1/3)

distance from (-1 , -1) to (0 , 1/3) = sqrt(1 + 4/3^2) = sqrt(13/9) = sqrt(13) / 3

distance from (-1 , -1) to (-1/13 , 5/13):

sqrt[(12/13)^2 + (18/13)^2] = sqrt(144 + 324) / 13 = 21.63 / 13 = 1.664

it's not by much, but (-1/13 , 5/13) is slightly closer to (-1 , -1) then (0 , 1/3)

the shortest distance from a point to a line is ALWAYS along the perpendicular line from that point to the line (any thing else would be the hypotenuse of a right triangle with the perpendicular distance as a leg, which MUST be shorter)

• the point which closest to the line is factor of intersection of the perpendicular drawn from (-4,a million) and the line. if the perpendicular is y=mx+c the position m=gradient and c is consistent m would properly be got here upon out with information from the formulation m1*m2= -a million given linei s 2x+3y-3=0------(a million) or 3y= -2x+3 or y= (-2/3)x+a million m1= -2/3 now m1*m2=-a million or (-2/3)*m2=-a million or m2=3/2 perpendicular's equation y=mx+c the following m= 3/2 y=a million x=-4 or a million=(3/2)*(-4)+c or a million= -6 +c or c=7 equation of perpendicur is y= 3/2 x+7 or 2y=3x+14 or -3x+2y=14 multiply with information from 2 you get -6x+4y=28---(3) and multiply with information from 3 the equation (a million) you get 6x+9y=9----(4) including (3) and (4) 13y=37 y= 37/13 putting this fee to (a million) sparkling up for 'x' 2x +3(37/13)=3 or 2x= 3-111/13=(39-111)/13= -seventy 2/13 so the nearest factor is ( -seventy 2/13,37/13) or ( -5.fifty 3,12.33) ans

• Find the point on the line 2x + 3y - 1 = 0, .................. 

which is closest to the point (-1, -1)?

x-intercept, x= 1/2,

y-intercept, y = 1/3

Thus the point closest to  is,

(0, 1/3).