The equation of the tangent to the curve $(1+x^2)y = 2-x$,where it crosses the $X$-axis,is

If $A = \{P(\alpha, \beta) \mid \text{the tangent drawn at } P \text{ to the curve } y^3 - 3xy + 2 = 0 \text{ is a horizontal line}\}$ and $B = \{Q(a, b) \mid \text{the tangent drawn at } Q \text{ to the curve } y^3 - 3xy + 2 = 0 \text{ is a vertical line}\}$, then $n(A) + n(B) = $

If the tangent to the curve $y=x^{3}-x^{2}+x$ at the point $(a, b)$ is also tangent to the curve $y=5x^{2}+2x-25$ at the point $(2, -1)$,then $|2a+9b|$ is equal to $........$

The angle of intersection of the curves $y^2 = \frac{2x}{\pi}$ and $y = \sin x$ is

On the curve $y=x^3$,the point at which the tangent line is parallel to the chord joining the points $(-1, -1)$ and $(2, 8)$ is

If the tangent drawn to the curve $(x^2+1)(y-3)=x$ at a point $P$,lying in the first quadrant,is a horizontal line,then the equation of the normal at the point $P$ is