The line which is parallel to the $x$-axis and crosses the curve $y = \sqrt{x}$ at an angle of $45^{\circ}$ is:

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

The area (in sq units) of the triangle formed by the normal drawn at the point $(1,0)$ on the curve $x=e^{\sin y}$ with the coordinate axes is

If the tangent to the curve $xy + ax + by = 0$ at $(1, 1)$ makes an angle $\tan^{-1}(2)$ with the $x$-axis,then $\left( \frac{a + b}{ab} \right)$ is equal to:

If the normal to the curve $y(x)=\int_{0}^{x}(2t^{2}-15t+10)dt$ at a point $(a, b)$ is parallel to the line $x+3y=-5$,where $a>1$,then the value of $|a+6b|$ is equal to..........

For the curve ${y^n} = {a^{n - 1}}x$,the subnormal at any point is constant. The value of $n$ must be