The correct option is D 1x+1z=1y
Given, QB⊥AC,PA⊥AC and RC⊥AC.
In ΔAPC and ΔBQC,
∠ACP=∠BCQ [common]
∠PAC=∠QBC [each 90∘]
∴ΔAPC∼ΔBQC [by AA similarity]
⇒APBQ=ACBC⇒xy=ACBC...(i)
Now, in ΔARC and ΔAQB,
∠ACR=∠ABQ [each 90∘]
∠CAR=∠BAQ [common]
⇒ΔARC∼ΔAQB [by AA similarity]
∴RCQB=ACAB
⇒zy=ACAB...(ii)
From Eqs. (i), we get
BC=yxAC ...(iii)
From Eq. (ii), we get
AB=yzAC
Adding Eqs. (iii) and (iv), we get
AB+BC=(yx+yz)AC⇒AC=y(1x+1z)AC⇒1y=1x+1z