A Troubling Problem!

The locus of the foot of perpendicular drawn from the center of the ellipse ${ x }^{ 2 }+{ 3y }^{ 2 }=6$ on any tangent to it is of the form ${ ({ x }^{ a }+{ y }^{ b }) }^{ c }=d{ x }^{ e }+f{ y }^{ g }$ , where $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , and $g$ are positive integers which are not necessarily different.

Find the value of $a+b+c+d+e+f+g$ .