Answer:
c) 6x + y = -52  is required equation perpendicular to the given equation.
Step-by-step explanation:
If the equation is of the form    : y = mx  + C.
Here m = slope of the equation.
Two equations are said to be perpendicular if the product of their respective slopes is -1.
Here, equation 1 :  -x + 6y = -12
or, 6y = -12  + x
or, y = (x/6)  - 2
⇒Slope of line 1 = (1/6)
Now, for equation 2  to be  perpendicular:
Check for each equation:
a. x + 6y = -67       ⇒  6y = -67  - x
or, y = (-x/6)  - (67/6)      ⇒Slope of line 2 = (-1/6)
but 
b. x - 6y = -52   ⇒  -6y = -52  - x
or, y = (x/6)  + (52/6)      ⇒Slope of line 2 = (1/6)
but 
c. 6x + y = -52     
or, y =y = -52  - 6x      ⇒Slope of line 2 = (-6)
 
Hence,  6x + y = -52  is required equation 2.
d. 6x - y = 52  ⇒  -y = 52  - 6x
or, y = 6x   - 52      ⇒Slope of line 2 = (6)
but 
Hence,  6x + y = -52  is  the  only required equation .