$\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{h}{a} + \frac{k}{b} =1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,given\,\,\,\,{a^2} + {b^2} = ab\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {ah + kb = 1}\limits_{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \frac{a}{b} + \frac{b}{a} = 1\\multiply\,\,\,{h^2} + {k^2} + hk(\frac{b}{a} + \frac{a}{b}) = 1\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{h^2} + {k^2} + hk = 1\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + {y^2} + xy - 1 = 0
\end{array}$
Note that the locus is not physically viable
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