well see another way is that .....



the locus of all the points from which the tangents to ellipse are mutually perpendicular is Director Circle of ellipse .



and equation of the director circle is {x}^{2} + {y}^{2} = {a}^{2} + {b}^{2}

since the point (a,3) are on the director circle ,

we have

{a}^{2} + 9 = 9 + 4

hence a=pm2