jmc

The parabola $y=a{x}^2+6$ is tangent to the line $y=x$ when the equation $$a{x}^2+6=x$$has a double root (which is the $x$-coordinate of the point of tangency). From this equation, $$a{x}^2-x+6=0.$$This quadratic has a double root when the discriminant is 0, which gives us $1-24a=0.$ Hence, $a=\boxed{\frac{1}{24}}.$