The length of the tangent drawn at the point $P\left(\frac{\pi}{4}\right)$ on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2^{\frac{2}{3}}$ is

The number of points on the curve $y=54x^5-135x^4-70x^3+180x^2+210x$ at which the normal lines are parallel to $x+90y+2=0$ is:

Let the curve be represented by $x=2(\cos t+t \sin t)$ and $y=2(\sin t-t \cos t)$. Then the normal at any point '$t$' of the curve is at a distance of . . . . . . units from the origin.

If the line $ax + by + c = 0$ is a normal to the curve $xy = 1$,then

If the lines $x+y=a$ and $x-y=b$ touch the curve $y = x^{2}-3x+2$ at the points where the curve intersects the $x$-axis,then $\frac{a}{b}$ is equal to:

Let $f: R \rightarrow R$ be a bijection. $A$ curve represented by $y=f(x)$ is such that $f^{\prime}(x)>0$ for all $x \in R$. The tangent and normal drawn at $P(\alpha, 1)$ on the curve cut the $X$-axis at $A$ and $B$ respectively, and $C$ is the foot of the perpendicular from $P$ onto the $X$-axis. If $P(\alpha, 1)$ is such a point that $AC+CB$ is minimum, then the tangent at $P$ is parallel to the line