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

The curve $y=ax^3+bx^2+cx+5$ touches the $X$-axis at $P(-2,0)$ and cuts the $Y$-axis at a point $Q$,where the gradient is $3$. Then the values of $a, b, c$ are

Let $f(x) = \begin{cases} -x^2 & \text{for } x < 0 \\ x^2 + 8 & \text{for } x \ge 0 \end{cases}$. Then the $x$-intercept of the line that is tangent to the graph of $f(x)$ is

For the curve $xy = c^2$,the subnormal at any point varies as

What is the length of the normal at any point on the curve $y = \frac{1}{2}a(e^{x/a} + e^{-x/a})$?

The angle between the curves $y^2=x$ and $x^2=y$ at the point $(1,1)$ is