The ratio of the length of the subnormal to the square of the length of the subtangent at any point $P$ on the curve $y^2=(2x+1)^3$ is

The length of the perpendicular drawn from the origin on the normal to the curve $x^2+2xy-3y^2=0$ at the point $(2,2)$ is

Prove that the curves $x=y^{2}$ and $xy=k$ cut at right angles if $8k^{2}=1$.

Let $S$ be the set of all natural numbers $n$ for which the line $\frac{x}{a} + \frac{y}{b} = 2$ is a tangent to the curve $\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$ at the point $(a, b)$,where $ab \neq 0$. Then:

The equation of the tangent to the curve $y = 2 \cos x$ at $x = \frac{\pi}{4}$ is

If the tangent drawn to the curve $y=x^3$ at a point $(\alpha, \beta)$ cuts the curve again at another point $(\alpha_1, \beta_1)$,then $\frac{\beta_1}{\beta}=$