Shortest distance

The distance of an arbitrary point (x,y) of the curve from (0,-3) is
d=\sqrt{x^2+(y+3)^2}=\sqrt{x^2+(4+a_1x^2+\ldots+ a_nx^{2n})^2}

since y=1+a_1x^2+a_2x^4+\ldots +a_nx^{2n}. Since all the a_i's are positive and the squares of real numbers are always non-negative, d will be minimum only when x=0. As such the shortest distance is 4.