If the tangent drawn at a point $P$ on the curve $y=3x^2-5x+7$ is parallel to its chord joining the points $(1, y_1)$ and $(2, y_2)$ on it, then the $x$-coordinate of the point $P$ 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

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

If the slope of the line through $(0,0)$ which is tangent to the curve $y=x^2+x+16$ is $m$,then the value of $m-4$ is

The equation of the normal to the curve $y^4=a x^3$ at $(a, a)$ is