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

Commented by mr 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).

findthecoordinatesoftheimageB′ofthepointB(p,q)inthemirrory=x2foranobserveratthepointA(h,k).

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

Commented by mr 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))

saythereflectionpointisP(t,t2)tanθ=y′=2ttanφ=q−t2p−ttanϕ=k−t2h−ttanα=tan(θ−φ)=2t−q−t2p−t1+2t×q−t2p−t=t2−2pt+q2t3−(2q−1)t−ptanα=tan(∅−θ)=k−t2h−t−2t1+k−t2h−t×2t=−t2−2ht+k2t3−(2k−1)t−h⇒t2−2pt+q2t3−(2q−1)t−p+t2−2ht+k2t3−(2k−1)t−h=0⇒t=....tangentlineatpointP:2tx−y−t2=0imageofB(p,q)abouttangentline:xB′=p−4t(2tp−q−t2)4t2+1yB′=q+2(2tp−q−t2)4t2+1

Commented by mr W last updated on 25/Jul/20

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