O(0,0), A(1,2), B(3,4) are the vertices of `DeltaOAB`. The joint equation of the altitude and median drawn from O is

Options

A. `x^(2) + 7xy - y^(2) = 0`
B. `x^(2) + 7xy + y^(2) = 0`
C. `3x^(2) -xy -2y^(2) = 0`
D. `3x^(2) + xy - 2y^(2) = 0`

Answer

D

Explanation

Given `O(0,0), A (1,2)` and `B(3,4)` be the vertices of `Delta OAB`.
Let OP and OD be the alitude and median of
`DeltaOAB`, respectively .
`:.` Coordinates of `D =` Mid of AB
`= ((1+3)/(2), (2+4)/(2)) = (4/2, 6/2) = (2,3)`
Now, equation of OD is `(y-0) = ((3-0)/(2-0)) (x-0)`
`[ :' y - y_(1) = ((y_(2)- y_(1))/(x_(2)-x_(1))(x-x_(1)))]`
`rArr y = 3/2 x rArr 2y = 3x rArr 3x - 2y = 0`
Slope of `OP = (-1)/("Slope of AB")` [ `:' OP _|_ AB`]
`= (-1)/(((3-1)/(4-2))) = (-1)/(2/2) =- 1`
`:.` Equation of OP is `(y - 0) = - 1(x-0)`
`[ :' y - y_(1) = "slope" (x-x_(1))]`
`rArr y = -x rArr x + y = 0`
Now, joint equation of OP and OD
`(x+y) (3x-2y) = 0`
`rArr 3x^(2) - 2xy + 3xy -2y^(2) = 0`
`rArr 3x^(2) + xy - 2y^(2) = 0`