(1,−2)=(α,−α−1) ⇒α=1 It is clear from question that one of the vertex of triangle is intersection of x-axis and x+y+1=0⇒A(−1,0) Let vertex B be (α,−α−1) Line AC⟂BH so, mAC⋅mBH=−1 ⇒O=−‌
(1−α)
α+2
⇒α=1⇒B(1,−2) Let vertex C be (β,0) Line AH⟂BC ∴mAH−mBC=−1 ⇒‌
1
2
⋅‌
2
β−1
=−1⇒‌‌β=0 Centroid of △ABC is (0,−‌
2
3
) We know that G (centroid) divides line joining circumcentre (O) and orthocentre (H) in the ratio 1:2.
2h+1=0⇒‌
2k+1
3
=−‌
2
3
⇒h=−‌
1
2
⇒k=−‌
3
2
⇒ Circumcentre is (−‌
1
2
,−‌
3
2
). Equation of circum circle is (passing through C(0,0) ) is x2+y2+x+3y=0