Tangents and it...

Equation of normal drawn to the graph of the function defined as $$f(x)=\dfrac{\sin\,x^2}{x},x \neq0 $$ and $$f(0)=0$$ at the origin is 

The slope of the tangent to the curve $$y=\int _{ 0 }^{ x }{ \frac { dt }{ 1+{ t }^{ 3 } }  } $$ at the point where $$x=1$$ is

A tangent PT is drawn to the circle $$x^2+y^2=4$$ at the point $$P(\sqrt{3}, 1)$$. A straight line L, perpendicular to PT is a tangent to the circle $$(x-3)^2+y^2=1$$. $$(1)$$ A possible equation of L is?

The points of the curve $$y={ x }^{ 3 }+x-2$$ at which its tangent are parallel to the straight line $$y=4x-1$$ are

The equation of the other normal to the parabola $${y}^{2}=4ax$$ which passes through the intersection of those at $$(4a,-4a)$$ and $$(9a,-6a)$$ is:

Let $$f\left( x \right) =\begin{cases} -{ x }^{ 2 }\ \ \ \ \ , x<0 \\ { x }^{ 2 }+8,x\ge 0 \end{cases}$$ Equation of tangent line touching both branches of $$y=f\left( x \right)$$ is

The area of the triangle formed by the positive x-axis, the tangent and normal to the curve $${ x }^{ 2 }+{ y }^{ 2 }=16{ a }^{ 2 }$$ at the point $$\left( 2\sqrt { 2 } a,2\sqrt { 2 } a \right) $$ is

A point P moves such that sum of the slopes of the normal drawn from it to the hyperbola $$xy=16$$ is equal to the sum of the ordinates of the feet of the normal. Let 'P' lies on the curve C, then.
The equation of 'C' is?

The equation of normal $${ x }^{ 2/3 }+{ y }^{ 2/3 }={ a }^{ 2/3 }$$ at $$\left( \cfrac { a }{ 2\sqrt { 2 }  } ,\cfrac { a }{ 2\sqrt { 2 }  }  \right) $$ is ______ .

The tangent to $$\left( a{ t }^{ 2 },2at \right) $$ is perpendicular to X-axis at _____ point $$t\in R$$.