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

If the curves $x=y^{4}$ and $xy=k$ cut at right angles,then $(4k)^{6}$ is equal to ..... .

If the tangent to the curve $2y^3 = ax^2 + x^3$ at the point $(a, a)$ cuts the coordinate axes at $p$ and $q$ such that $p^2 + q^2 = 61$,then what is the value of $a$?

Let $f(x) = \begin{cases} -x^2 & \text{for } x < 0 \\ x^2 + 8 & \text{for } x \ge 0 \end{cases}$. Then the $x$-intercept of the line that is tangent to the graph of $f(x)$ is

The sum of the lengths of the subtangent and the subnormal drawn at $\theta = \frac{\pi}{3}$ on the cycloid $x = a(\theta - \sin \theta)$,$y = a(1 - \cos \theta)$ is

If the tangent drawn at $A(2,1)$ to the curve $x=1+\frac{1}{y^2}$ meets the curve again at $B$,then