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Question
Eliminate θ, if
x = 3 cosec θ + 4 cot θ
y = 4 cosec θ – 3 cot θ
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Solution
Given:
x = 3cosecθ + 4cotθ .....(1)
y = 4cosecθ – 3cotθ .....(2)
Multiplying (1) by 4 and (2) by 3, we get
4x = 12cosecθ + 16cotθ .....(3)
3y = 12cosecθ – 9cotθ .....(4)
Subtracting (4) from (3), we get
4x − 3y = 25cot θ
⇒ cot2θ = \[\left( \frac{4x - 3y}{25} \right)^2\] .....(5)
Multiplying (1) by 3 and (2) by 4, we get
3x = 9cosecθ + 12cotθ .....(6)
4y = 16cosecθ – 12cotθ .....(7)
Adding (6) and (7), we get
3x + 4y = 25cosecθ
⇒ cosecθ = \[\frac{3x + 4y}{25}\]
⇒ cosec2θ = \[\left(\frac{3x + 4y}{25}\right)^2\] .....(8)
\[{cosec}^2 \theta - \cot^2 \theta = \left( \frac{3x + 4y}{25} \right)^2 - \left( \frac{4x - 3y}{25} \right)^2 = 1\]
\[ \Rightarrow \left( \frac{3x + 4y}{25} \right)^2 - \left( \frac{4x - 3y}{25} \right)^2 = 1\]
\[ \Rightarrow \frac{1}{{25}^2}\left[ \left( 3x + 4y \right)^2 - \left( 4x - 3y \right)^2 \right] = 1\]
\[ \Rightarrow \left( 3x + 4y \right)^2 - \left( 4x - 3y \right)^2 = 625\]
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