If the normal to the curve $y = f(x)$ at the point $(3, 4)$ makes an angle of $3\pi / 4$ with the positive $X$-axis,find $f'(3)$.

At which point on the curve $y = x^2 + 3x$ should the tangent be drawn so that it passes through the point $(0, -9)$?

The chord of the curve $y=x^{2}+2ax+b$ joining the points where $x=\alpha$ and $x=\beta$ is parallel to the tangent to the curve at abscissa $x$ equal to:

Find the points on the curve $9y^2 = x^3$ where the normal to the curve makes equal intercepts with the axes.

An equation of the tangent to the curve $y = x^4$ from the point $(2, 0)$ not on the curve is

Find the point on the curve $x = a(\theta + \sin \theta)$,$y = a(1 - \cos \theta)$ where the tangent makes an angle of $\pi / 4$ with the $x$-axis.