If $y=4x-5$ is a tangent to the curve $y^2=px^3+q$ at $(2,3)$,then $p-q$ 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:

Let the curve be represented by $x=2(\cos t+t \sin t)$ and $y=2(\sin t-t \cos t)$. Then the normal at any point '$t$' of the curve is at a distance of . . . . . . units from the origin.

If the lines $x+y=a$ and $x-y=b$ touch the curve $y = x^{2}-3x+2$ at the points where the curve intersects the $x$-axis,then $\frac{a}{b}$ is equal to:

The two curves $x=y^2$ and $xy=a^3$ cut orthogonally at a point,then $a^2$ is equal to

Find the slope of the tangents to the curve $y = (x + 1)(x - 3)$ at the points where it meets the $x$-axis.