Find the equation of the tangent line to the curve $y=x^{2}-2x+7$ which is perpendicular to the line $5y-15x=13$.

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.

Let $\ell$ be a line which is normal to the curve $y=2x^2+x+2$ at a point $P$ on the curve. If the point $Q(6,4)$ lies on the line $\ell$ and $O$ is the origin,then the area of the triangle $OPQ$ is equal to.......

The area of the triangle formed by the normal to the curve $y = e^{2x} + x^2$ at the point $(0, 1)$ with the coordinate axes is $......$ square units.

Let $y=f(x)$ be any curve on the $X-Y$ plane and $P$ be a point on the curve. Let $C$ be a fixed point not on the curve. If the length $PC$ is either a maximum or a minimum, then:

Find the equation of the normal to the curve $y = \frac{x-7}{(x-2)(x-3)}$ at the point where it cuts the $X$-axis.