$A(-2,9)$ and $B(1,6)$ are two points on the curve $y=x^2+5$. The coordinates of the point $C$ on the curve such that the tangent drawn at $A$ is parallel to the chord $BC$,are

The subnormal at any point on the curve $xy^n = a^{n + 1}$ is constant for $n=$ ............

$P(5,2)$ is a point on the curve $y=f(x)$ and $\frac{7}{2}$ is the slope of the tangent to the curve at $P$. The area of the triangle (in sq. units) formed by the tangent and the normal to the curve at $P$ with the $x$-axis is:

The line $x + y = 0$ touches the curve $y^2 = ax^3 + b$ at the point $(1, -1)$. Find the values of $a$ and $b$.

If the curves $y = \frac{\ln x}{x}$ and $y = \lambda x^2$ (where $\lambda$ is a constant) touch each other,then $\lambda$ is

The distance,from the origin,of the normal to the curve,$x = 2\cos t + 2t\sin t, y = 2\sin t - 2t\cos t$ at $t = \frac{\pi}{4}$,is