If l cosec
+ m cot + n = 0 and
l' cosec + m' cot + n' = 0. Show that: (mn'-m'n)^2 - (nl'-n'l)^2 = (lm'-l'm)^2.
Answers
Step-by-step explanation:
\boxed{\mathsf{ Solution : }}
Solution:
\mathsf{ Given, }Given,
\begin{gathered}\mathsf{ \implies l \: cosec \: \theta \: + \: m \: cot \: \theta \: + \:n \: = \: 0 \qquad...(1) \: } \\ \\ \mathsf{ \implies l' \: cosec \: \theta \: + \: m' \: cot \: \theta \: + \: n' \: = 0 \qquad...(2) } \\ \\ \mathsf{ Using \: Cross \: Multiplication \: Method, } \end{gathered}
⟹lcosecθ+mcotθ+n=0...(1)
⟹l
′
cosecθ+m
′
cotθ+n
′
=0...(2)
UsingCrossMultiplicationMethod,
\begin{gathered}\mathsf{ Coe. \: of \: cosec \: \theta \quad Coe. \: of \: cot \: \theta \quad Const. \: term } < br / > \\ \\ \mathsf{ \qquad \: l \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \qquad \qquad m \qquad \qquad n } \\ \\ \mathsf{ \: \qquad l' \qquad \qquad \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: m' \qquad \qquad n' } \end{gathered}
Coe.ofcosecθCoe.ofcotθConst.term<br/>
lmn
l
′
m
′
n
′
\mathsf{ Now, }Now,
\begin{gathered} \\ \mathsf{ \implies \dfrac{cosec\: \theta}{mn' \: - \:m'n } \: = \: \dfrac{ cot \: \theta}{nl' \: - \:n'l} \: = \: \dfrac{1}{lm' \: - \:l'm } } \end{gathered}
⟹
mn
′
−m
′
n
cosecθ
=
nl
′
−n
′
l
cotθ
=
lm
′
−l
′
m
1
\mathsf{ Now, }Now,
\begin{gathered} \\ \mathsf{ \implies \dfrac{ cosec \: \theta }{mn' \: - \: m'n } \: = \: \dfrac{1}{lm' \: - \: l'm}} < br / > \\ \\ \\ \mathsf{ \therefore \: cosec\: \theta \: = \: \dfrac{ mn' \: - \: m'n \: }{ lm' \: - \: l'm } \qquad...(3)} \end{gathered}
⟹
mn
′
−m
′
n
cosecθ
=
lm
′
−l
′
m
1
<br/>
∴cosecθ=
lm
′
−l
′
m
mn
′
−m
′
n
...(3)
\begin{gathered}\mathsf{ Now, } \\ \\ \\ \mathsf{ \implies \dfrac{ cot \: \theta }{nl' \: - \:n'l} \: = \: \dfrac{1}{lm' \: - \: l'm }} \\ \\ \\ \mathsf{ \implies \: cot \: \theta \: = \: \dfrac{ nl' \: - \: n'l }{ lm' \: - \: l'm }}\end{gathered}
Now,
⟹
nl
′
−n
′
l
cotθ
=
lm
′
−l
′
m
1
⟹cotθ=
lm
′
−l
′
m
nl
′
−n
′
l
\begin{gathered}\mathsf{ Using \: trigonometric \: identity , } \\ \\ \\ < br / > < br / > < br / > \boxed{\mathsf{\implies cot \: \theta \: = \: \sqrt{ \: {cosec}^{2} \: \theta \: - \: 1 \: }}} < br / > \end{gathered}
Usingtrigonometricidentity,
<br/><br/><br/>
⟹cotθ=
cosec
2
θ−1
<br/>
\mathsf{\implies \sqrt{\: {cosec}^{2} \: \theta \: - \: 1 }\: = \: \dfrac{nl' \: - \: n'l}{lm' \: - \: l'm}} < br / >⟹
cosec
2
θ−1
=
lm
′
−l
′
m
nl
′
−n
′
l
<br/>
< br / > \mathsf{ \implies {cosec}^{2} \: \theta \: - \: 1 \: = \: {(\dfrac{nl' \: - \:n'l\:}{lm' \: - \:l'm \:})}^{2}}<br/>⟹cosec
2
θ−1=(
lm
′
−l
′
m
nl
′
−n
′
l
)
2
\mathsf{ Plug \: the \: value \: of \:(3), }Plugthevalueof(3),
\begin{gathered} \mathsf{\implies {(\dfrac{mn' \: - \:m'n \:}{lm' \: - \:l'm})}^{2} \: - \:1 \: = \:{(\dfrac{nl' \: - \:n'l\:}{lm' \: - \:l'm \:})}^{2}} \\ \\ \end{gathered}
⟹(
lm
′
−l
′
m
mn
′
−m
′
n
)
2
−1=(
lm
′
−l
′
m
nl
′
−n
′
l
)
2
\begin{gathered}\mathsf{ \implies \dfrac{{( \:mn' \: - \:m'n \:)}^{2}}{{( \: lm' \: - \: l'm \: )}^{2}} \: - \: 1 \: = \: \dfrac{{( \:nl' \: - \:n'l \: )}^{2}}{{( \: lm' \: - \:l'm \:)}^{2}}} < br / > \\ \\ \end{gathered}
⟹
(lm
′
−l
′
m)
2
(mn
′
−m
′
n)
2
−1=
(lm
′
−l
′
m)
2
(nl
′
−n
′
l)
2
<br/>
\begin{gathered}\mathsf{ \implies \dfrac{{ ( mn' \: - \: m'n \: )}^{2} \: - \: {( \: lm' \: - \: l'm \: )}^{2} }{ \cancel{{( \: lm' \: - \: l'm \: )}^{2}} } \: = \: \dfrac{{( \:nl' \: - \:n'l \: )}^{2}}{ \cancel{{( \: lm' \: - \:l'm \:)}^{2}}}} \\ \\ \end{gathered}
⟹
(lm
′
−l
′
m)
2
(mn
′
−m
′
n)
2
−(lm
′
−l
′
m)
2
=
(lm
′
−l
′
m)
2
(nl
′
−n
′
l)
2
\underline{\mathsf{ \therefore \: { ( \: mn' \: - \: m'n \: )}^{2} \: - \: {( \: nl'\: - \: n'l \: )}^{2} \: = \: { ( \: lm' \: - \: l'm \: )}^{2}}} < br / >
∴(mn
′
−m
′
n)
2
−(nl
′
−n
′
l)
2
=(lm
′
−l
′
m)
2
<br/>