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 `DeltaOAB`. Let OP and OD be the altitude and median off `DeltaOAB`, respectively.
`therefore`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)`
`[becausey-y_(1)=((y_(2)-y_(1))/I(x_(2)-x_(1))(x-x_(1)))]`
`impliesy=(3)/(2)ximplies2y=3ximplies3x-2y=0`
Slope of `OP=(-1)/("Slope of AB")` [`becausePObotAB`]
`=(-1)/(((3-1)/(4-2)))=(-1)/((2)/(2))=-1`
`therefore` Equation of OP is (y-0)=-1(x-0)
`[becausey-y_(1)="slope"(x-x_(1))]`
`impliesy=-x`
`impliesx+y=0`
Now, joint equationn of OP and OD
`(x+y)(3x-2y)=0`
`3x^(2)-2xy+3xy-2y^(2)=0`
`implies3x^(2)+xy-2y^(2)=0`