If the points of contact of the tangents drawn from $(0,0)$ to the curve $y=x^2+3x+4$ are $(\alpha, \beta)$ and $(\gamma, \delta)$,then $\beta+\delta=$

The sum of the lengths of the subtangent and the subnormal drawn at $\theta = \frac{\pi}{3}$ on the cycloid $x = a(\theta - \sin \theta)$,$y = a(1 - \cos \theta)$ is

$A$ straight line $L$ passing through the point $(2,3)$ bounds the minimum area with the curve $f(x) = x^2 - 4x + 6$. Find the equation of the tangent to the curve parallel to $L$.

If the tangent and the normal drawn to the curve $xy^2 + x^2y = 12$ at the point $(1, 3)$ meet the $X$-axis in $T$ and $N$ respectively, then $TN =$

If the tangent to the curve $y = f(x) = x \log_{e} x$ $(x > 0)$ at a point $(c, f(c))$ is parallel to the line segment joining the points $(1, 0)$ and $(e, e)$,then $c$ is equal to:

If $y=4x-5$ is tangent to the curve $y^2=px^3+q$ at $(2,3)$,then