The given lines are and x+3y‌=10 . . . (i) and 6x2+xy−y2‌=0
or ‌‌6x2+3xy−2xy−y2=0 or ‌‌3x(2x+y)−y(2x+y)=0 3x−y=0 . . . (ii) 2x+y=0 . . . (iii) On solving Eqs. (i) and (ii), we get ⇒‌x+3⋅(3x)‌=10 ⇒‌10x‌=10 ‌‌ and ‌‌x‌=1 ⇒‌3⋅1−y‌=0 ‌‌ Coordinates of ‌B‌ are ‌(1,3)‌ On solving Eqs. (ii) and (iii), we get x=0,y=0 ∴ Coordinates of A are (0,0). A line perpendicular to BC is 3x−y=λ It passes through (0,0), then 0−0=λ ⇒‌‌λ=0 The line AD is ‌‌3x−y=0 . . . (iv) A line perpendicular to AC is x−2y=λ It passes through (1,3), then 1−6=λ λ=−5 The line BE is ‌‌x−2y=−5 . . . (v) On solving Eqs. (iv) and (v), we get x=1,y=3 Thus, the coordinates of required orthocentre is (1,3).