Let a conic $$C$$ pass through the point $$(4, -2)$$ and $$P(x, y), x \geq 3$$, be any point on $$C$$. Let the slope of the line touching the conic $$C$$ only at a single point $$P$$ be half the slope of the line joining the points $$P$$ and $$(3, -5)$$. If the focal distance of the point $$(7, 1)$$ on $$C$$ is $$d$$, then $$12d$$ equals ______

We are given that the conic $$C$$ passes through $$(4, -2)$$, and for any point $$P(x, y)$$ on $$C$$ with $$x \geq 3$$, the slope of the tangent at $$P$$ equals half the slope of the line joining $$P$$ and $$(3, -5)$$.

The slope of the tangent at $$P$$ is $$\frac{dy}{dx}$$.

The slope of the line joining $$P(x, y)$$ and $$(3, -5)$$ is $$\frac{y - (-5)}{x - 3} = \frac{y + 5}{x - 3}$$.

The given condition is:

$$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{y + 5}{x - 3}$$

This is a separable ODE:

$$\frac{dy}{y + 5} = \frac{dx}{2(x - 3)}$$

Integrating both sides:

$$\ln|y + 5| = \frac{1}{2}\ln|x - 3| + \ln K$$

$$|y + 5| = K\sqrt{|x - 3|}$$

Since $$x \geq 3$$, we have $$|x - 3| = x - 3$$. So:

$$y + 5 = K\sqrt{x - 3}$$ (taking the appropriate sign)

Using the point $$(4, -2)$$: $$-2 + 5 = K\sqrt{4 - 3}$$, so $$3 = K \cdot 1$$, giving $$K = 3$$.

The conic is $$y + 5 = 3\sqrt{x - 3}$$, or equivalently:

$$(y + 5)^2 = 9(x - 3)$$

This is a parabola with vertex at $$(3, -5)$$ and axis parallel to the x-axis.

The standard form is $$(y - k)^2 = 4a(x - h)$$ where the vertex is $$(h, k)$$ and $$4a = 9$$, so $$a = \frac{9}{4}$$.

The focus is at $$(h + a, k) = \left(3 + \frac{9}{4}, -5\right) = \left(\frac{21}{4}, -5\right)$$.

The directrix is $$x = h - a = 3 - \frac{9}{4} = \frac{3}{4}$$.

The focal distance of a point $$(x_0, y_0)$$ on a parabola is the distance from the point to the focus, which equals $$x_0 - h + a = x_0 - 3 + \frac{9}{4}$$.

For the point $$(7, 1)$$:

First verify it lies on $$C$$: $$(1 + 5)^2 = 36$$ and $$9(7 - 3) = 36$$. Yes, $$36 = 36$$ ✓.

$$d = x_0 - 3 + \frac{9}{4} = 7 - 3 + \frac{9}{4} = 4 + \frac{9}{4} = \frac{25}{4}$$

Therefore:

$$12d = 12 \times \frac{25}{4} = 75$$

The answer is $$75$$.