The value of r is :

It is given that PK and QR are parallel, mPK = mQR ⇒ 2at𕒴at2𕒶a = 2at′𕒶ara(t′)2−ar2 ⇒ tt2𕒶 = t′−r(t′)2−r2Also, astt′ = 𕒵, on cross multiplication, we get −t′−tr2 = −t−rt2−2t′+2r ⇒ −tr2+r(t2−2)+t′+t=0The roots of this quadratic equation in r are α, β = −(t2𕒶)±√(t2𕒶)2𕒸(−t(t+t′))𕒶t ⇒ α, β = (2−t2)±√t4𕒶t ⇒ α, β = (2−t2)±t2𕒶t ⇒ α = 𕒵t, β = (t2𕒵)tBut, α cannot be possible as R and Q will be the same as t′ = 𕒵t. Hence the required value of r is (t2𕒵)t.