Let P and Q be any two points on the lines represented by `2x-3y = 0` and `2x + 3y = 0` respectively. If the area of triangle OPQ (where O is origin) is 5, then which of the following is not the possible equation of the locus of mid-point of PQ?

Options

A. A. `4x^(2) - 9y^(2) +30 = 0`
B. B. `4x^(2) - 9y^(2) - 30 = 0`
C. C. `9x^(2) - 4y^(2) - 30 = 0`
D. D. none of these

Answer

C

Explanation

We have
Area `(DeltaOPQ) = (1)/(2) |quad{:(0,0,1),(a,(2a)/(3),1),(b,(-2b)/(3),1):}|=5` (Given)
`rArr (4ab)/(3) = +-10`
So `4ab = +- 30` (i)
Also `2h = a+b` (ii)
and `2k = (2a-2b)/(3)`or `a - b =3k` (iii)
As `4ab = (a+b)^(2) -(a-b)^(2)`
`rArr +- 30 = 4h^(2) - 9k^(2)` [Using (i),(ii) and (iii)]
So required locus can be `4x^(2) - 9y^(2) = +30`.