The length of the subtangent at any point $(x_1, y_1)$ on the curve $y=5^x$ is

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 coordinates of the point on the curve $y=x^2-3x+2$ where the tangent is perpendicular to the straight line $y=x$ are

The area of the triangle formed by the coordinate axes and a tangent to the curve $xy = a^2$ at the point $(x_1, y_1)$ is . . . . . . sq. units (where $a, x_1$,and $y_1$ are non-zero).

The perpendicular distance from the origin to the normal drawn at any point on the curve $x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta)$ is

Let $M$ and $N$ be the number of points on the curve $y^{5}-9xy+2x=0$,where the tangents to the curve are parallel to the $x$-axis and $y$-axis,respectively. Then the value of $M + N$ equals $......$