The tangent to a given curve $y=f(x)$ is perpendicular to the $x$-axis,if

The equations of straight lines which are both tangent and normal to the curve $27x^2 = 4y^3$ are

The tangent to the curve $y = e^x$ at the point $(c, e^c)$ intersects the line joining the points $(c - 1, e^{c-1})$ and $(c + 1, e^{c+1})$ at a point whose $x$-coordinate is:

The length of the tangent to the curve $x = a(\cos t + \log \tan(t/2)), y = a \sin t$ is

The equation of the tangent to the curve $y=x^3-2x+7$ at the point $(1,6)$ is

If the length of the subnormal at any point on the curve $y = a^{1-n}x^n$ is constant,then $n = \dots$