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Q.
The locus of the point of intersection of the lines $ x\,cot\,\theta +y\,cosec\,\theta \text{=}2 $ and $ x\,cosec\,\theta +y\,cot\,\theta =6 $ is
J & K CETJ & K CET 2003
Solution:
Given equations are $ x\,\,\cot \,\theta +y\,\text{cosec }\theta \text{=2} $ ...(i) and $ \text{x cosec }\theta +y\,\,\cot \,\theta =6 $ ...(ii) On squaring and subtracting Eq. (i) from Eq. (ii), we get
$ {{x}^{2}}(\text{cose}{{\text{c}}^{2}}\theta -{{\cot }^{2}}\theta )+{{y}^{2}}({{\cot }^{2}}\theta -\text{cose}{{\text{c}}^{2}}\theta ) $
$={{(6)}^{2}}-{{(2)}^{2}} $
$ \Rightarrow $ $ {{x}^{2}}-{{y}^{2}}=32 $
It represents an equation of hyperbola.