The tangent and normal to the curve $y = x^2 - x + 4$ at $P(1, 4)$ intersect the $X$-axis at $A$ and $B$ respectively. Then the area of $\Delta PAB$ is .......... square units.

If $f: R \rightarrow R$ is a function defined for all $x \in R$ by $f(x)=x^3+f^{\prime}(1) x^2+f^{\prime \prime}(2) x-f^{\prime \prime \prime}(3)$,then the area (in sq. units) of the triangle formed by the $X$-axis,the tangent,and the normal drawn to the curve $y=f(x)$ at $x=0$ is

What are the lengths of the intercepts on the coordinate axes made by the tangent to the curve $y = 2x^2 + 3x - 2$ at the point $(1, 3)$?

The equation of the tangent to the curve $y = \sin x$ at the point $(\pi, 0)$ is $......$.

The equation of the normal to the curve $2x^{2} + y^{2} = 12$ at the point $(2, 2)$ is

If a normal drawn at a point $P$ to the curve $y=\sin x$ passes through the origin,then the locus of $P$ is