The normal to the curve $y=f(x)$ at the point $(3,4)$ makes an angle $\frac{3 \pi}{4}$ with the positive $X$-axis. Then $f^{\prime}(3)$ is equal to:

If the tangent to the curve $2y^3 = ax^2 + x^3$ at the point $(a, a)$ cuts off intercepts $\alpha$ and $\beta$ on the coordinate axes,where $\alpha^2 + \beta^2 = 61$,then the value of $|a|$ is

If $\theta$ is an angle between the curves $x^2+4y=0$ and $xy=2$,then $\tan \theta=$

The angle of intersection of the curves $r = \sin \theta + \cos \theta$ and $r = 2 \sin \theta$ is equal to:

If $\beta$ is an angle between the normals drawn to the curve $x^2+3y^2=9$ at the points $P(3 \cos \theta, \sqrt{3} \sin \theta)$ and $Q(-3 \sin \theta, \sqrt{3} \cos \theta)$,where $\theta \in (0, \frac{\pi}{2})$,then:

If the curves $2x = y^2$ and $2xy = K$ intersect perpendicularly,then the value of $K^2$ is