Statement I : The locus of the point, whose distance from the x-axis is twice its distance from the $\mathrm{y}\text{-axis is }{\mathrm{y}}^2=4{\mathrm{x}}^2$
Statement II : The locus of the point $(\cot \mathrm{\theta }+\cos \mathrm{\theta },\cot \mathrm{\theta }-\cos \mathrm{\theta })\text{ is }{\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)}^2=16\mathrm{xy}$ which of the above are true.

Statement I :- The locus of the point, whose distance from the x-axis is twice its distance from the y-axis  so,

       $$Distance from the x-axis is $\left|y\right|$

Distance from the y-axis is $\left|x\right|$ so,

$$\begin{gathered} \left|y\right|=2\left|x\right| \\ s.o.b.s \\ {y}^2=4x^2 \end{gathered}$$

$\begin{array}{r}\mathrm{h}=\cot \theta +\cos \theta \dots \text{ (i) }\\ \mathrm{k}=\cot \theta -\cos \theta \dots \text{ (ii) }\end{array}$

Eq (i) + Eq (ii)

$\left(\frac{\mathrm{h}+\mathrm{k}}{2}\right)=\cot \theta$   ... (iii)

Eq (i) - Eq (ii)

$\left(\frac{\mathrm{h}-\mathrm{k}}{2}\right)=\cos \theta$

From Eq (iii)

$$\begin{gathered} {\left(\frac{\mathrm{h}+\mathrm{k}}{2}\right)}^2=\frac{{\cos }^2\theta }{{\sin }^2\theta }=\frac{{\cos }^2\theta }{1-{\cos }^2\theta } \\ \Rightarrow {\left(\frac{\mathrm{h}+\mathrm{k}}{2}\right)}^2=\frac{(\mathrm{h}-\mathrm{k}{)}^2}{4-(\mathrm{h}-\mathrm{k}{)}^2} \\ \Rightarrow \left[4-(\mathrm{h}-\mathrm{k}{)}^2\right]\times (\mathrm{h}+\mathrm{k}{)}^2=4(\mathrm{h}-\mathrm{k}{)}^2 \\ \Rightarrow 4\left[{\mathrm{h}}^2+{\mathrm{k}}^2+2\mathrm{hk}-{\mathrm{h}}^2-{\mathrm{k}}^2+2\mathrm{hk}\right]={\left({\mathrm{h}}^2-{\mathrm{k}}^2\right)}^2 \\ \Rightarrow 16\mathrm{hk}={\left({\mathrm{h}}^2-{\mathrm{k}}^2\right)}^2 \\ \Rightarrow {\left({\mathrm{h}}^2-{\mathrm{k}}^2\right)}^2=16\mathrm{hk} \\ \Rightarrow {\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)}^2=16\mathrm{xy} \end{gathered}$$