Tangents and it...

At what point the tangent line to the curve $$\displaystyle y=\cos \left ( x+y \right ),\left ( -2\pi \leq x\leq 2\pi  \right )$$ is parallel to $$x + 2y = 0$$

The line $$\dfrac x a +\dfrac  y  b = 1$$ touches the curve $$\displaystyle y=be^{-\tfrac xa}$$ at the point -

At what values of $$a$$, the curve $$x^4+3ax^3+6x^2+5$$ is not situated below any of its tangent lines

The point at which the tangent to the curve $$\displaystyle y=x^{3}+5$$ is perpendicular to the line $$x + 3y = 2$$ are

The points on the curve $$\displaystyle y^{2}=4a\left ( x+a\sin \frac{x}{a} \right )$$ at which the tangent is parallel to x axis lie on -

The equation of normal to the curve $$x+y=x^{y}$$, where it cuts x-axis is

The lines tangent to the curve $$x^3-y^3+x^2y-yx^2+3x-2y=0$$ and $$x^5-y^4+2x+3y=0$$ at the origin intersect at an angle $$\theta$$ equal to

The points on the curve $$\displaystyle 9y^2=x^{3}$$ where the normal to the curve makes equal intercepts with coordinates axes is :

If the line $$x -y = 0$$ is tangent to $$f(x) = b \ln x - x$$, then $$b$$ lies in the interval

Let tangent at a point P on the curve $$\displaystyle { x }^{ 2m }={ y }^{ \tfrac { n }{ 2 }  }={ a }^{ \tfrac { 4m+n }{ 2 }  }$$ meets the x-axis and y-axis at A and B respectively, If AP:PB is $$\displaystyle \frac { n }{ \lambda m } $$, where P lies between A and B, then find the value of $$\displaystyle \lambda $$