If the line $y=4x-5$ touches the curve $y^2=ax^3+b$ at the point $(2,3)$,then $7a+2b=$

If $T$ is the length of the subtangent drawn at any point on the curve $3 y^2 = 4 x^3$ and $N$ is the length of the subnormal at the same point,then $(\beta T)^2 =$

If $ST$ and $SN$ are the lengths of the subtangent and the subnormal at the point $\theta = \frac{\pi}{2}$ on the curve $x = a(\theta + \sin \theta ), y = a(1 - \cos \theta ), a \neq 0$,then:

If the tangent to the curve $y=x+\sin y$ at a point $(a, b)$ is parallel to the line joining $\left(0, \frac{3}{2}\right)$ and $\left(\frac{1}{2}, 2\right),$ then

What are the lengths of the intercepts on the coordinate axes made by the tangent to the curve $y = 2x^2 + 3x - 2$ at the point $(1, 3)$?

The equation of the normal to the curve $y(1+x^{2})=2-x$ at the point where the tangent crosses the $x$-axis is