Tangents and it...

The points of contact of the vertical tangents $$x= 2-3\sin \theta $$, $$y= 3+2\cos \theta $$ are

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

Equation of the line through the point $$(1/2,2)$$ and tangent to the parabola $$\displaystyle y=\frac{-x^{2}}{2}+2$$ and secant to the curve $$\displaystyle y=\sqrt{4-x^{2}}$$ is

If the curve $$\displaystyle { \left( \frac { x }{ a }  \right)  }^{ n }+{ \left( \frac { y }{ b }  \right)  }^{ n }=2$$ touches the straight line $$\displaystyle \frac { x }{ a } +\frac { y }{ b } =2$$, then find the value of $$n$$.

Find all the tangents to the curve $$\displaystyle y=\cos \left ( x+y \right ),-2\pi \leq \times \leq 2\pi $$ that are parallel to the line $$x + 2y = 0$$

A function is defined parametrically by the equations
x= $$\displaystyle 2t+t^{2}\sin \frac{1}{t}$$   if $$\displaystyle t\neq 0$$;    $$ 0  $$,   otherwise
  and 

The curve $$\displaystyle y=ax^{3}+bx^{2}+cx+5$$ touches the $$x$$ - axis at $$P(-2, 0)$$ and cuts the $$y$$-axis at a point $$Q$$, where its gradient is $$3$$. Find $$a, b, c$$.

Let $$f$$ be a continuous, differentiable and bijective function. If the tangent to $$y=f\left( x \right) $$ at $$x=b$$, then there exists at least one $$c\in \left( a,b \right) $$ such that 

Find the equation of the normal to the curve $$\displaystyle y = \left ( 1+x \right )^{y}+\sin ^{-1}\left ( \sin ^{2}x \right )$$ at $$x = 0$$

Find the equation of normal to the curve $$\displaystyle x^{2}=4y$$ passing through the point $$(1, 2)$$