The normal at the point $(1,1)$ on the curve $2y + x^{2} = 3$ is

If the normal to the curve $y = f(x)$ at the point $(4, 6)$ makes an angle $\frac{2\pi}{3}$ with the positive $x$-axis in the anticlockwise direction,then $f'(4)$ is:

The area (in sq. units) of the triangle formed by the tangent and normal drawn to the curve $(\frac{x}{3})^n+(\frac{y}{4})^n=2$ at $(3,4)$ and the $X$-axis is

If the equation of the tangent at $(2,3)$ on the curve $y^2 = ax^3 + b$ is $y = 4x - 5$,then the value of $a^2 + b^2$ is:

If $y=2x$ is a tangent to the curve $y^2=ax^3+b$ at $(1,2)$,then $(a, b)=$

$x_1, x_2 \in N$. If a line having slope $2$ is a tangent to the curve $y=x^4-6x^3+13x^2-10x+5$ at points $P(x_1, y_1)$ and $Q(x_2, y_2)$,then $x_1x_2+y_1y_2=$