The point on the curve $y=x^2+4x+3$ which is closest to the line $y=3x+2$ is

The number of points on the curve $y=2t^2+3t-5$ and $x=t^3-4t^2-3t$ such that the normals drawn at them on the curve are parallel to the $X$-axis is

The coordinates of the point $P$ on the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$, where the tangent is inclined at an angle $\frac{\pi}{4}$ to the $x$-axis, are

The curve $\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2$,where $n \in N$,touches the line at the point $(a, b)$. Find the equation of the line.

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 point$(s)$ at which the tangents to the curve $y = x^3 - 3x^2 - 7x + 6$ cut off a line segment on the positive semi-axis $OX$ that is half the length of the segment cut off on the negative semi-axis $OY$ are given by: