Find the points on the curve $\frac{x^{2}}{4} + \frac{y^{2}}{25} = 1$ at which the tangents are parallel to the $x$-axis.

Find the equation of the tangent to the curve $y = \sqrt{3x - 2}$ which is parallel to the line $4x - 2y + 5 = 0$.

If the length of the subnormal at any point of the curve $y^n = a^{n-1}x$ is constant,then $n$ equals-

Find the equation of the tangent line to the curve $y=x^{2}-2x+7$ which is parallel to the line $2x-y+9=0$.

If the tangent at the point $(x_{1}, y_{1})$ on the curve $y=x^{3}+3x^{2}+5$ passes through the origin,then $(x_{1}, y_{1})$ does $NOT$ lie on which of the following curves?

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