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The locus of a point, from where the tangents to the rectangular hyperbola \( x^{2}-y^{2}=a^{2} \) contain an angle of \( 45^{\circ} \),
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(1) \( \left(x^{2}+y^{2}\right)^{2}+a^{2}\left(x^{2}-y^{2}\right)=4 a^{2} \)
(2) \( 2\left(x^{2}+y^{2}\right)^{2}+4 a^{2}\left(x^{2}-y^{2}\right)=4 a^{2} \)
(3) \( \left(x^{2}+y^{2}\right)^{2}+4 a^{2}\left(x^{2}-y^{2}\right)=4 a^{4} \)
(4) \( \left(x^{2}+y^{2}\right)^{2}+a^{2}\left(x^{2}-y^{2}\right)=a^{4} \)
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