If the curves, $$x^2 - 6x + y^2 + 8 = 0$$ and $$x^2 - 8y + y^2 + 16 - k = 0$$, $$(k \gt 0)$$ touch each other at a point, then the largest value of $$k$$ is ___________.

We have two curves

$$x^{2}-6x+y^{2}+8=0$$

and

$$x^{2}-8y+y^{2}+16-k=0,\qquad k\gt 0.$$

Each of these equations represents a circle. To see their centres and radii we first write each circle in standard form by completing the square.

For the first curve, isolate the terms in $$x$$:

$$x^{2}-6x = (x-3)^{2}-9.$$

Substituting this into the first equation gives

$$\bigl[(x-3)^{2}-9\bigr] + y^{2}+8 = 0.$$

Simplifying,

$$(x-3)^{2}+y^{2} -1 = 0,$$

or equivalently,

$$(x-3)^{2}+y^{2}=1.$$

Thus the first circle has centre

$$C_{1} = (3,\,0)$$

and radius

$$r_{1} = 1.$$

Now we treat the second curve. Complete the square in $$y$$:

$$y^{2}-8y = (y-4)^{2}-16.$$

Substituting back,

$$x^{2} + \bigl[(y-4)^{2}-16\bigr] + 16 - k = 0.$$

The constants $$-16$$ and $$+16$$ cancel, so we obtain

$$x^{2} + (y-4)^{2} - k = 0,$$

or

$$x^{2} + (y-4)^{2} = k.$$

Hence the second circle has centre

$$C_{2} = (0,\,4)$$

and radius

$$r_{2} = \sqrt{k}.$$

When two circles touch each other, the distance $$d$$ between their centres satisfies one of the following standard conditions:

• External tangency: $$d = r_{1} + r_{2}.$$

• Internal tangency: $$d = |\,r_{1} - r_{2}\,|.$$

First we compute the centre-centre distance. Using the distance formula,

$$d = \sqrt{(3-0)^{2} + (0-4)^{2}} = \sqrt{9 + 16} = 5.$$

Now we analyse the two possible tangencies.

External tangency. Here

$$r_{1} + r_{2} = d \;\Longrightarrow\; 1 + \sqrt{k} = 5.$$

So

$$\sqrt{k} = 4 \;\Longrightarrow\; k = 16.$$

Internal tangency. Here

$$|\,r_{1} - r_{2}\,| = d.$$

Because $$r_{2} \gt r_{1}$$ (otherwise the left side would be negative), we write

$$r_{2} - r_{1} = 5 \;\Longrightarrow\; \sqrt{k} - 1 = 5.$$

Therefore

$$\sqrt{k} = 6 \;\Longrightarrow\; k = 36.$$

Between the two admissible values $$k = 16$$ and $$k = 36$$, the question asks for the largest value. Clearly,

$$k_{\max} = 36.$$

Hence, the correct answer is Option D.