Draw PL′ parallel OM and PL parallel ON
In △PLMcosω=LMPL
⇒LM=PLcosω=kcosωOM=OL+LMOM=h+kcosω........(i)
In △PL′Ncosω=L′NPL′
⇒L′N=PL′cosω=hcosωON=OL′+L′NON=k+hcosω......(ii)
Given OM+ON=2c
using (i) and (ii)
h+kcosω+k+hcosω=2ch+k+cosω(h+k)=2c(h+k)(1+cosω)=2c(h+k)(1+2cos2ω2−1)=2ch+k=ccos2ω2h+k=csec2ω2
Replacing h by x and k by y
x+y=csec2ω2
is the required locus of P