Question Number 148901 by mr W last updated on 01/Aug/21

Commented bymr W last updated on 01/Aug/21

answer to Q148806

Commented byTawa11 last updated on 01/Aug/21

Weldone sir.

Commented byTawa11 last updated on 01/Aug/21

Am not the one that asked the question sir. I just appreciate your effort.

Commented bymr W last updated on 04/Aug/21

plane (x/a)+(y/b)+(z/c)=1 intersects the coordinate axes at A,B,C. A(a,0,0) B(0,b,0) A(0,0,c) center of circumcircle of ΔABC is D. D(α,β,γ) (α−a)^2 +β^2 +γ^2 =r^2 α^2 +(β−b)^2 +γ^2 =r^2 α^2 +β^2 +(γ−c)^2 =r^2 a^2 −2aα+α^2 +β^2 +γ^2 =r^2 with s^2 =α^2 +β^2 +γ^2 α=((a^2 −r^2 +s^2 )/(2a)) β=((b^2 −r^2 +s^2 )/(2b)) γ=((c^2 −r^2 +s^2 )/(2c)) s^2 =(((a^2 −r^2 +s^2 )/(2a)))^2 +(((b^2 −r^2 +s^2 )/(2b)))^2 +(((c^2 −r^2 +s^2 )/(2c)))^2 4s^2 =(((a^2 −r^2 )/a))^2 +(((b^2 −r^2 )/b))^2 +(((c^2 −r^2 )/c))^2 +((1/a^2 )+(1/b^2 )+(1/c^2 ))s^4 .... radius of circle is r. r=AB=((√(a^2 +b^2 +c^2 ))/2) OD^(→) =((a/2),(b/2),(c/2)) normal of plane n^(→) =((1/a),(1/b),(1/c)) say u^(→) ,v^(→) are two perpendicular unit vectors in the plane. OP^(→) =OD^(→) +DP^(→) =OD^(→) +r cos θ u^(→) +r sin θ v^(→) u^→ is ⊥ n^(→) , u_x ×(1/a)+u_y ×(1/b)+u_z ×(1/c)=0 for example u_x =a, u_y =−b, u_z =0 i.e. u^(→) // AB^(→) v^→ =n^(→) ×u^(→) = determinant (((1/a),((1/b) ),(1/c)),(a,(−b),0))=((b/c),(a/c),−(a/b)−(b/a)) since u^(→) , v^(→) are unit vectors, u^→ =((a/( (√(a^2 +b^2 )))),−(b/( (√(a^2 +b^2 )))),0) v^→ =((b/( c(√((a^2 +b^2 )((1/a^2 )+(1/b^2 )+(1/c^2 )))))),(a/( c(√((a^2 +b^2 )((1/a^2 )+(1/b^2 )+(1/c^2 )))))),−(((a^2 +b^2 ))/( ab(√((a^2 +b^2 )((1/a^2 )+(1/b^2 )+(1/c^2 ))))))) OP^(→) =((a/2),(b/2),(c/2))+((√(a^2 +b^2 +c^2 ))/(2(√(a^2 +b^2 ))))(a,−b, 0) cos θ+((√(a^2 +b^2 +c^2 ))/(2(√((a^2 +b^2 )((1/a^2 )+(1/b^2 )+(1/c^2 )))))) ((b/c),(a/c),−((a^2 +b^2 )/(ab)))sin θ or { ((x=(a/2)+((a(√(a^2 +b^2 +c^2 )))/(2(√(a^2 +b^2 ))))cos θ+((b(√(a^2 +b^2 +c^2 )))/(2c(√((a^2 +b^2 )((1/a^2 )+(1/b^2 )+(1/c^2 )))))) sin θ)),((y=(b/2)−((b(√(a^2 +b^2 +c^2 )))/(2(√(a^2 +b^2 ))))cos θ+((a(√(a^2 +b^2 +c^2 )))/(2c(√((a^2 +b^2 )((1/a^2 )+(1/b^2 )+(1/c^2 )))))) sin θ)),((z=(c/2)−(((a^2 +b^2 )(√(a^2 +b^2 +c^2 )))/(2ab(√((a^2 +b^2 )((1/a^2 )+(1/b^2 )+(1/c^2 )))))) sin θ)) :} this is the equation of circumcircle with θ as parameter.