$\text{ If the locus of the moving point }P(x,y)\text{ satisfying }\sqrt{(x-1{)}^2+y^2}+\sqrt{(x+1{)}^2+(y-\sqrt{12}{)}^2}=a\text{ is ellipse, }$

$\text{ then the least integral }v\text{alue of }a\text{ is }$

$\begin{array}{c}\text{ We have }\sqrt{(x-1{)}^2+y^2}+\sqrt{(x+1{)}^2+(y-\sqrt{12}{)}^2}=a\\ \text{ distance between }\text{ distance between }\\ P(x,y)\text{ and }S(1,0)P^{\mathrm{\prime }}(x,y)\text{ and }{S}^{\mathrm{\prime }}(-1,\sqrt{12})\end{array}$

$\therefore SP+S^{\mathrm{\prime }}P=a$

$\text{ So, locus of }P\text{ is ellipse if }$

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}a>S{S}^{\mathrm{\prime }}\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}a>\sqrt{4+12}\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}a>4\end{array}$