Tangents and it...

The abscissa of the points, where the tangent to curve $$y={x}^{3} - 3{x}^{2} - 9x+5$$ is parallel to x-axis, are

Suppose that the equation $$f\left( x \right) ={ x }^{ 2 }+bx+c=0$$ has two distinct real roots $$\alpha $$ and $$\beta $$. The angle between the tangent to the curve $$y=f\left( x \right) $$ at the point $$\left( \dfrac { \alpha +\beta  }{ 2 } ,f\left( \dfrac { \alpha +\beta  }{ 2 }  \right)  \right) $$ and the positive direction of the $$x$$-axis is

The points on the curve $$9{y}^{2}={x}^{3}$$, where the normal to the curve makes equal intercepts with the axes are

The coordinates of the point P on the curve $$x = a(\theta + \sin \theta), y = a(1 - \cos \theta)$$ where the tangent is inclined at an angle $$\dfrac {\pi}{4}$$ to the x-axis, are

Slope of Normal to the curve $$y = x^{2} - \dfrac {1}{x^{2}}$$ at $$(-1, 0)$$ is

The slope of the tangent to the curve $$x=3t^2+1, y=t^3-1$$ at $$x=1$$ is 

The equation of one of the curves whose slope at any point is equal to $$y+2x$$ is

The slope of the tangent to the curve $$x={t}^{2}+3t-8$$, $$y=2{t}^{2}-2t-5$$ at the point $$(2,-1)$$ is

If $$\Delta$$ is the area of the triangle formed by the positive x-axis and the normal and tangent to the circle $$x^{2} + y^{2} = 4$$ at $$(1, \sqrt {3})$$, then $$\Delta =$$

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 equal to