If in a parallelogram ABCD, the coordinate of A, B and C are respectively (1,2), (3,4) and (2,5), then the equation of the diagonal BD is:- 5x+3y-1=0, 3x-y+7=0, 3x+5y-3=0, 5x-y+9=0

Let the coordinates of D be (x, y).
In a parallelogram, the diagonals bisect each other.
Midpoint of AC = ( (1+2)/2, (2+5)/2 ) = (3/2, 7/2)
Midpoint of BD = ( (3+x)/2, (4+y)/2 )
Since midpoints are the same, we have:
(3+x)/2 = 3/2 => 3+x = 3 => x = 0
(4+y)/2 = 7/2 => 4+y = 7 => y = 3
Therefore, the coordinates of D are (0, 3).
Now let's check which equation is satisfied by (0,3) and (3,4):
5x + 3y - 1 = 0
5(0) + 3(3) - 1 = 8 ≠ 0
3x - y + 7 = 0
3(0) - 3 + 7 = 4 ≠ 0
3x + 5y - 3 = 0
3(0) + 5(3) - 3 = 12 ≠ 0
5x - y + 9 = 0
5(0) - 3 + 9 = 6 ≠ 0
Let's use the slope of BD. The slope of AC is (5-2)/(2-1) = 3. Since BD is a diagonal, it must be perpendicular to AC.
Slope of BD = (4-3)/(3-0) = 1/3. This is incorrect (should be -1/3)
There must be a mistake in the question or the options. Let's re-examine the midpoint calculation:
Midpoint of AC = ((1+2)/2, (2+5)/2) = (1.5, 3.5)
Midpoint of BD = ((3+x)/2, (4+y)/2)
(3+x)/2 = 1.5 => 3+x = 3 => x = 0
(4+y)/2 = 3.5 => 4+y = 7 => y = 3
So D = (0,3). Let's check the options again:
5x + 3y - 1 = 0: 5(0) + 3(3) - 1 = 8 ≠ 0
3x - y + 7 = 0: 3(0) - 3 + 7 = 4 ≠ 0
3x + 5y - 3 = 0: 3(0) + 5(3) - 3 = 12 ≠ 0
5x - y + 9 = 0: 5(0) - 3 + 9 = 6 ≠ 0
None of the options seem to be correct. There might be an error in the problem statement or the given options.