If the lines 4x + 3y = 84 and 3x + ky + 7 = 0 are perpendicular to each other. Then the value of k is :

1. 4

2. -4

3. $\dfrac{1}{4}$

4. $-\dfrac{1}{4}$

Given,

1st equation :

⇒ 4x + 3y = 84

⇒ 3y = -4x + 84

⇒ y = $-\dfrac{4}{3}x + \dfrac{84}{3}$

⇒ y = $-\dfrac{4}{3}x$ + 28

Comparing above equation with y = mx + c, we get :

Slope of first equation = $-\dfrac{4}{3}$

2nd equation :

⇒ 3x + ky + 7 = 0

⇒ ky = -3x - 7

⇒ y = $-\dfrac{3}{k}x - \dfrac{7}{k}$

Slope of second equation = $-\dfrac{3}{k}$

We know that,

Product of slope of two perpendicular lines = -1.

$\Rightarrow -\dfrac{4}{3} \times -\dfrac{3}{k} = -1 \\[1em] \Rightarrow \dfrac{4}{k} = -1 \\[1em] \Rightarrow -k = 4 \\[1em] \Rightarrow k = -4.$

Hence, Option 2 is the correct option.