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

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 $y^2 = x$ does the tangent make an angle of $45^{\circ}$ with the $x-$axis?

If the slope of the tangent drawn at any point $(x, y)$ to the curve $y=f(x)$ is $3x^2-5$ and $f(1)=2$, then the tangent at $(1, 2)$ to the curve $y=f(x)$ intersects the curve at the point

The equation of the tangent to the curve $y=\sqrt{9-2x^2}$ at the point where the ordinate and abscissa are equal is:

The ratio of the length of the subnormal to the square of the length of the subtangent at any point $P$ on the curve $y^2=(2x+1)^3$ is