Question

If l sec theta - m tan theta + n = 0 and l' sec theta - m' tan theta + n' = 0 , then prove that :

Answer 1

lsecθ-mtanθ+n=0
or, lsecθ-mtanθ=-n
l'secθ-m'tanθ+n'=0
or, l'secθ-m'tanθ=-n'
∴, (mn'-m'n)²-(ln'-nl')²
=[m{-(l'secθ+m'tanθ)}-m'{-(lsecθ+mtanθ)}]²-[{-(l'secθ+m'tanθ)}l-{-(lsecθ+mtanθ)}l']²
=(-ml'secθ-mm'tanθ+m'lsecθ+mm'tanθ)²-(-ll'secθ-lm'tanθ+ll'secθ+ml'tanθ)²
=sec²θ(m'l-ml')²-tan²θ(ml'-lm')²
=sec²θ(ml'-lm')²-tan²θ(ml'-lm')²
=(ml'-lm')²(sec²θ-tan²θ)
=(ml'-lm')² [∵, sec²Ф-tan²Ф=1]
=(ml'-lm')² (Proved)

Answer 2

by cross multiplication find Tan and sec

Use identity