Question

Find the value of k so that 48x-ky=11 and (k+2)x+16y=-19 are perpendicular lines.

Answer 1

Answer: k = -1 +/- √769

Explanation:

48x - ky = 11

-48x -48x

-ky = -48x + 11

`(-ky)/(-k) = (-48x)/(-k) + (11)/(-k)`

`y =(48x)/(k) - (11)/(k)`

Slope:
`(48)/(k)`

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(k + 2)x + 16y = -19

- (k + 2)x -(k + 2)x

16y = -(k + 2)x - 19

`(16y)/(16) = -((k + 2)x)/(16) - (19)/(16)`

`y = -((k + 2)x)/(16) - (19)/(16)`

Slope:
`-((k + 2))/(16)`

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`(48)/(k)` and
`-((k + 2))/(16)` are perpendicular so they have opposite signs and are reciprocals of each other. When multiplied by its reciprocal, their product equals -1.

`-((k + 2))/(16)` *
`(k)/(48)` = -1

`((k + 2)k)/(16(48))` = 1

Cross multiply, then solve for the variable.

(k + 2)(k) = 16(48)

k² + 2k - 768 = 0

Use quadratic formula to solve:

k = -1 +/- √769