If the curves, x² – 6x + y² + 8 = 0 and x² – 8y + y² + 16 – k = 0, (k > 0) touch each other at a point, then the largest value of k is ______.

The equation of the two curves in this question are similar to the equation for circles in a cartesian plane: for a circle of radius [tex]r[/tex] centered at [tex](a,\, b)[/tex], the standard equation of the circle would be:

[tex](x - a)^{2} + (y - b)^{2} = r^{2}[/tex].

Expand to obtain:

[tex]x^{2} - 2\, a\, x + a^{2} + y^{2} - 2\, b\, y + b^{2} = r^{2}[/tex].

[tex]x^{2} - 2\, a\, x + y^{2} - 2\, b\, y + (a^{2} + b^{2} - r^{2}) = 0[/tex].

Rearrange [tex]x^{2} - 6\, x + y^{2} + 8 = 0[/tex] to obtain a standard equation of the form [tex](x - a)^{2} + (y - b)^{2} = r^{2}[/tex]:

[tex]x^{2} - 6\, x + y^{2} + 8 = 0[/tex].

[tex](x^{2} - 6\, x + 9) + y^{2} - 1 = 0[/tex].

[tex](x - 3)^{2} + (y - 0)^{2} = 1^{2}[/tex].

In other words, this equation represents a circle of radius [tex]r_{1} = 1[/tex] centered at [tex](3,\, 0)[/tex].

Similarly, rearrange to obtain a standard equation for [tex]x^{2} - 8\, y + y^{2} + 16 - k = 0[/tex]:

[tex]x^{2} + (y^{2} - 8\, y + 16) = k[/tex].

[tex]\displaystyle (x - 0)^{2} + (y - 4)^{2} = \left(\sqrt{k}\right)^{2}[/tex].

In other words, this equation represents a circle of radius [tex]r_{2} = \sqrt{k}[/tex] centered at [tex](0,\, 4)[/tex].

The distance between the center of the two circles is [tex]\displaystyle d = \sqrt{3^{2} + 4^{2}} = 5[/tex].

The center of both circle are fixed. The radius [tex]r_{1}[/tex] of the circle centered at [tex](3,\, 0)[/tex] is also fixed. There are a number of possible [tex]r_{2}[/tex] values that would allow these two circles would touch each other at exactly one point.

Refer to the diagram attached (not to scale):

The two circles might not overlap each other. Sum of the radius of the two circles is equation to the distance between the two centers: [tex]r_{1} + r_{2} = d[/tex], such that [tex]k = (r_{2})^{2} = (d - r_{1})^{2} = 16[/tex].
Alternatively, one of the two circles might completely engulfs the other. Difference between the radius of the two circle would be equal to the distance between the two centers: [tex]r_{2} - r_{1} = d[/tex], such that [tex]k = (r_{2})^{2} = (d + r_{1})^{2} = 36[/tex].

Hence, the largest possible real value of [tex]k[/tex] would be [tex]36[/tex].