Since, 0(0,0), A(1,2) and B(3,4) are the vertices of ΔOAB. Consider that OP and OD are altitude and median of ΔOAB, respectively. Then, Coordinates of D =(
1+3
2
,
2+4
2
)=(2,3) So, equation of OD is (y−0)= (
3−0
2−0
)(x−0) Hence, y =
3
2
x ⇒ 3x - 2y = 0 Now, slope of OP =
−1
SlopeofAB
−1
(
3−1
4−2
)
=−1[As,OP⊥AB] =
1
(
3−1
4−2
)
=−1 ∴ Equation of OP is (y - 0) = -1 (x - 0) ∴ y = - x ⇒ x + y = 0 Hence, joint equation of OP and OD is: (x + y)(3x - 2y) = 0 ⇒ 3x2+xy+y2 = 0