Find a value for k so that the line through (k, 6) and (4, 2) is perpendicular to the line with equation x + 2y = 2.Application

The correct answer is: Value of k for the given line is 6.

Step-by-step solution:-
x + 2y = 2
∴ 2y = -x + 2
∴ y = -1/2 x + 1 ................................................ (Dividing both sides by 2)
Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = -1/2 ......................... (Equation i)
The given line is perpendicular to x + 2y = 2 and we know that perpendicular lines have slopes that are negative reciprocals of each other.
∴ Slope of the given line = -1/ slope of line (x + 2y = 2)
∴ Slope of the given line = m = -1/ -1/2 = 2 ..................... (From Equation i) .................................................................... (Equation ii)
The given line passes through the point (k,6) & (4,2).
Hence, x1 = k, y1 = 6, x2 = 4 & y2 = 2
(y-y1) = (y2-y1) × (x-x1)
(x2-x1)
∴ (y-6) = (2-6) × (x-k)
4-k
∴ y - 6 = -4 × (x - k)
4-k
∴ (y-6) × (4-k) = -4 ×(x-k)
∴ 4y - ky - 24 + 6k = -4x + 4k
∴ 4y - ky = -4x + 4k - 6k + 24
∴ (4-k) y = -4x -2k + 24
∴ y = -4/(4-k) x - 2/(4-k) k + 24/(4-k) ................................................. (Equation iii)
Comparing Equation iii with the standard form of a straight line i.e. y = mx + c, we get-
m = - 4/(4-k)
∴ 2 = - 4/(4-k) ..................................... (From Equation ii)
∴ 2 (4-k) = - 4 ...................................... [(Multiplying both sides by (4-k)]
∴ 8 - 2k = - 4
∴ -2k = - 4 - 8
∴ -2k = -12
∴ k = 6 .............................................. (Dividing both sides by -2)