If the curves $$y^2 = 6x$$, $$9x^2 + by^2 = 16$$ intersect each other at right angles, then the value of b is:

We have two curves, the parabola $$y^{2}=6x$$ and the conic $$9x^{2}+by^{2}=16$$. To find the condition for them to cut at right angles we shall first obtain the slopes of their tangents (i.e. $$\dfrac{dy}{dx}$$) at any common point and then use the fact that for two curves to intersect orthogonally, the product of their slopes at the point of intersection must be $$-1$$.

Step 1 : Slope of the tangent to the parabola.
Starting with $$y^{2}=6x$$, we differentiate implicitly with respect to $$x$$. Using the rule $$\dfrac{d}{dx}(y^{2})=2y\dfrac{dy}{dx}$$ and $$\dfrac{d}{dx}(6x)=6$$, we get $$2y\dfrac{dy}{dx}=6.$$ Solving for $$\dfrac{dy}{dx}$$, $$\dfrac{dy}{dx}=\dfrac{6}{2y}=\dfrac{3}{y}.$$ So the slope of the tangent to the first curve is $$m_{1}=\dfrac{3}{y}.$$

Step 2 : Slope of the tangent to the second curve.
For $$9x^{2}+by^{2}=16$$ we again differentiate implicitly with respect to $$x$$. Using $$\dfrac{d}{dx}(9x^{2})=18x$$ and $$\dfrac{d}{dx}(by^{2})=2by\dfrac{dy}{dx}$$, we have $$18x+2by\dfrac{dy}{dx}=0.$$ Isolating $$\dfrac{dy}{dx}$$ gives $$2by\dfrac{dy}{dx}=-18x$$ $$\dfrac{dy}{dx}=\dfrac{-18x}{2by}=\dfrac{-9x}{by}.$$ Thus the slope of the tangent to the second curve is $$m_{2}=\dfrac{-9x}{by}.$$

Step 3 : Condition for orthogonality.
For two curves to meet at right angles we must have $$m_{1}\,m_{2}=-1.$$ Substituting the expressions of $$m_{1}$$ and $$m_{2}$$ obtained above, $$\left(\dfrac{3}{y}\right)\left(\dfrac{-9x}{by}\right)=-1.$$ Multiplying the numerators and denominators, $$\dfrac{-27x}{by^{2}}=-1.$$ Cancelling the negative sign on both sides, $$\dfrac{27x}{by^{2}}=1.$$ Cross-multiplying, $$27x=by^{2}.$$

Step 4 : Eliminating $$y^{2}$$ using the first curve.
From the parabola we already have $$y^{2}=6x.$$ Substituting this value of $$y^{2}$$ into the relation $$27x=by^{2}$$, we obtain $$27x=b(6x).$$ Dividing both sides by $$x$$ (the intersection point cannot have $$x=0$$ because that would force $$y=0$$, which does not satisfy the ellipse equation), $$27=6b.$$ Finally, solving for $$b$$ gives $$b=\dfrac{27}{6}=\dfrac{9}{2}.$$

Hence, the correct answer is Option A.