Question Number 104769 by mr W last updated on 23/Jul/20

Commented bymr W last updated on 23/Jul/20

find the coordinates of the image B′   of the point B(p,q) in the mirror  y=x^2  for an observer at the point  A(h,k).

Answered by mr W last updated on 24/Jul/20

Commented bymr W last updated on 25/Jul/20

say the reflection point is P(t,t^2 )  tan θ=y′=2t  tan ϕ=((q−t^2 )/(p−t))  tan φ=((k−t^2 )/(h−t))  tan α=tan (θ−ϕ)=((2t−((q−t^2 )/(p−t)))/(1+2t×((q−t^2 )/(p−t))))=((t^2 −2pt+q)/(2t^3 −(2q−1)t−p))  tan α=tan (∅−θ)=((((k−t^2 )/(h−t))−2t)/(1+((k−t^2 )/(h−t))×2t))=−((t^2 −2ht+k)/(2t^3 −(2k−1)t−h))  ⇒((t^2 −2pt+q)/(2t^3 −(2q−1)t−p))+((t^2 −2ht+k)/(2t^3 −(2k−1)t−h))=0  ⇒t=....    tangent line at point P:  2tx−y−t^2 =0    image of B(p,q) about tangent line:  x_(B′) =p−((4t(2tp−q−t^2 ))/(4t^2 +1))  y_(B′) =q+((2(2tp−q−t^2 ))/(4t^2 +1))

Commented bymr W last updated on 25/Jul/20