Perp. Coordinate problem

[karthikpandian19]

The co-ordinates of the foot of the perpendicular from the point (2, 2) on the line y - 2x - 8 = 0 are

(A) (-1, 6)

(B) (-1, -4)

(C) (-2, 10)

(D) (2, 4)

(E) (-2, 4)

[aneesh.kg]

Slope of the line y - 2x = 8 is 2.

The slope the perpendicular drawn to it will be -(1/2).
(because product of slopes of perpendicular lines is -1)


Let's find the slope of the line with slope = -(1/2) and passing through (2,2)
2(y - 2) = -(x - 2)
2y + x = 6

The point of intersection of the two lines will be the foot of the perpendicular.
Solving them, we get (x, y) = (-2,4)

[spoiler](E)[/spoiler] is the answer

[GmatKiss]

Looking for a better explanation

[dabral]

Slightly different approach.

Dabral

[neelgandham]

Let me try(just try!)

Equation of the line is y - 2x - 8 = 0. For the point (x1,y1) to be on the line y - 2x - 8 = 0, the value of (y1 -2*x1 - 8) should be equal to zero. i.e.
y1 -2*x1 - 8 = 0

(A) (-1, 6) = y1 -2*x1 - 8 = 6 -(2*-1) - 8 = 8-8 = 0, Can be the answer
(B) (-1, -4) = y1 -2*x1 - 8 = -4 -(2*-1) - 8 = -10 !=0, It is definitely not the answer
(C) (-2, 10) = y1 -2*x1 - 8 = 10 -(2*-2) - 8 = 6 !=0, It is definitely not the answer
(D) (2, 4) = y1 -2*x1 - 8 = 4 -(2*2) - 8 = -8!=0, It is definitely not the answer
(E) (-2, 4) = = y1 -2*x1 - 8 = 4 -(2*-2) - 8 = 8-8 = 0, Can be the answer

Now it is between A and E. Since, the slope of the line y - 2x - 8 is 2, the slope of the line perpendicular to the same line is -1/2.

(A) Slope of line joining (2,2) and (-1,6) = 6-2/-1-2 = -4/3.It is definitely not the answer
(E) Slope of line joining (2,2) and (-2,4) = 4-2/-2-2 = -1/2. Bingo ! So, answer choice E is the correct answer

p.s: In my opinion, the solution by aneesh.kg(and Dabral) is a better and an easier one.
p.p.s: Dabral : Do you use Wacom Bamboo for the drawings ??