The length of the subnormal to any point on a curve is always constant. Then,the curve is . . . . . . .

Find the equation of tangents to the curve $y = \cos(x + y)$,$-2\pi \leq x \leq 2\pi$ that are parallel to the line $x + 2y = 0$.

If the tangent at the point $(x_{1}, y_{1})$ on the curve $y=x^{3}+3x^{2}+5$ passes through the origin,then $(x_{1}, y_{1})$ does $NOT$ lie on which of the following curves?

Let $M$ and $N$ be the number of points on the curve $y^{5}-9xy+2x=0$,where the tangents to the curve are parallel to the $x$-axis and $y$-axis,respectively. Then the value of $M + N$ equals $......$

The area of the triangle formed by the coordinate axes and the normal to the curve $y = e^{2x} + x^2$ at the point $x = 0$.

The length of the segment of the tangent line to the curve $x=a \cos ^3 t, y=a \sin ^3 t$, at any point on the curve cut off by the coordinate axes is