If the curve $y=ax^2-6x+b$ passes through $(0,4)$ and has its tangent parallel to the $x$-axis at $x=\frac{3}{2}$,then the values of $a$ and $b$ respectively are

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

$y=f(x)$ and $x=g(y)$ are two curves and $P(x, y)$ is a common point of the two curves. If at $P$, on the curve $y=f(x)$, $\frac{dy}{dx}=Q(x)$ and at the same point $P$ on the curve $x=g(y)$, $\frac{dx}{dy}=-Q(x)$, then

If $x^2 + y^2 = R^2$ $(R > 0)$,then $k = \frac{y''}{\sqrt{(1 + (y')^2)^3}}$. The value of $k$ in terms of $R$ alone is equal to:

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

Let $P(h, k)$ be a point on the curve $y=x^{2}+7x+2$ nearest to the line $y=3x-3$. Then the equation of the normal to the curve at $P$ is: