find the separate equation ofx²+2(cosec α)xy+y²=0

the given homogeneous equation of 2 degree is $x^2+2.\cos ec\alpha .xy+y^2=0$ ...........(1)
$$\begin{gathered} x^2+\left[\right(\cos ec\alpha +cot\alpha )+(\cos ec\alpha -cot\alpha \left)\right]xy+y^2=0 \\ {x}^2+(\cos ec\alpha +cot\alpha ).xy+(\cos ec\alpha -cot\alpha )xy+y^2=0 \\ x.[x+(\cos ec\alpha +cot\alpha \left)y\right]+y.(\cos ec\alpha -cot\alpha ).[x+\frac{y}{\cos ec\alpha -cot\alpha }]=0 \\ x.[x+(\cos ec\alpha +cot\alpha \left)y\right]+y.(\cos ec\alpha -cot\alpha ).\left[x+y.(\cos ec\alpha +cot\alpha \right]=0 \\ [\sin ce \frac{1}{\cos ec\alpha -cot\alpha }=\frac{1}{\cos ec\alpha -cot\alpha }*\frac{\cos ec\alpha +cot\alpha }{\cos ec\alpha +cot\alpha }=\frac{\cos ec\alpha +cot\alpha }{\cos e{c}^2\alpha -co{t}^2\alpha } \\ \Rightarrow [x+y.(\cos ec\alpha +cot\alpha )\left]\right[x+y.(\cos ec\alpha -cot\alpha ]=0 \\ \Rightarrow x+y.(\cos ec\alpha +cot\alpha )=0 or \\ x+y.(\cos ec\alpha -cot\alpha )=0 \end{gathered}$$

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