(r,s) will lie on the line with equation y = 3x + 2, if in place of x and y respectively we put the coordinates of the point and the line equation is satisfied.
So , (r,s) will lie on y = 3x +2, if s=3r+2 or in other words:
3r-s+2 = 0.
From St. 1:
(3r + 2 - s)(4r + 9 - s) = 0
i.e.
either
(3r + 2 - s) = 0 or (4r + 9 - s) = 0.
Say (3r + 2 - s) = 0 -->St. 1 satisfied and (r, s) lies on given line.
Now say, (3r + 2 - s) not equal to zero, but
(4r + 9 - s) = 0
Then St. 1 satisfied but (r, s) does not lie on given line.
So St. 1 is not sufficient.
Exactly same logic for St. 2.
St. 2 not sufficient.
Combining both statements.
case 1: (3r + 2 - s) is equal to zero
case 2: (3r + 2 - s) is not equal to zero
let's discuss case (2):
This means that
(4r + 9 - s) = 0 and also (4r - 6 - s) = 0
which is absurd. (4r-s = -9 as well as 6--> not possible)
so only case 1 holds i.e.
(3r + 2 - s) is equal to zero
(C) is the answer.
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Thanks & Regards,
Anaira Mitch