Question

If p and q are the lengths of perpendiculars from the origin to the
lines x cose - ysin 0 = k cos 20 and x sec 0 + y cosec 0 =k, respectively
that p? + 4q2 = k.​

Answer 1

Step-by-step explanation:

(x,y)= (0,0)

d=| Ax1+By1+C|÷√A²+√B²

P=| Cos ø×0- Sin ø×0-K Cos2ø|÷√Cos²ø+√Sin²ø

=|-K Cos2ø|= K Cos2ø

Q= xSecø+ yCosecø- K=0

Q=| Secø×0+ Cosecø×0- K|÷√Sec²ø+√Cosec²ø

Q= |-K |÷√1/Cos²ø+1/Sin²ø

= K ÷√Sin²ø + √Cos²÷√Sin²ø Cos²ø

= K÷√1÷Sin²ø Cos²ø

= K Sinø Cosø

p²+ 4q²= K²

LHS

K²Cos²2ø+ 4K²Sin²ø Cos²ø

K²(1-Sin²2ø)+ 4K²Sin²ø Cos ²ø

K²-K²Sin²2ø+ 4K²Sin²ø Cos ²ø

K²-K(2Sinø Cosø)²+ 4K²Sin²øCos²ø

K²- K²(4Sin²ø Cos²ø)+ 4K²Sin²øCos²ø

K²-4K²Sin²ø Cos²ø+ 4K²Sin²ø Cos ²ø

= K²= RHS