Tangents and it...

The gradient of the tangent line at the point $$(a cos \alpha, a sin \alpha)$$ to the circle $$x^2 + y^2 = a^2$$, is

If the product of the slope of tangent to curve at $$(x,y)$$ and its y-co-ordinate is equal to the x-co-ordinate of the point, then it represent.

The length of the segment of the tangent line to the curve $$x=a\cos ^{ 3 }{ t } ,y=\sin ^{ 3 }{ t } $$, at any point on the curve cut off by the coordinate axes is

Which one of the following be the gradient of the hyperbola $$xy=1$$ at the point $$\left(t,\dfrac{1}{t}\right)$$

Equation of tangent at that point of the curve $$y = 1 - {e^{\frac{x}{2}}}$$, where it meets y-axis  

The slope of the tangent to the curve $$xy+ax-by=0$$ at the point $$(1,1)$$ is $$2$$, then value of $$a$$ and $$b$$ are respectively:

The Equation of the tangent to the curves $${y^2} = 8x$$ and $$xy = -1$$ is

If the tangent to the curve $$y=x\log { x } $$ at $$\left( c,f\left( x \right)  \right) $$ is parallel to the line-segment joining $$A\left(1,0\right)$$ and $$B\left(e,e\right)$$, then c=...... .

The values of $$x$$ for which the tangents to the curves $$y=x\cos{x},y=\cfrac{\sin{x}}{x}$$ are parallel to the axis of $$x$$ are roots of  (respectively)

A curve with equation of the form $$y=a{x}^{4}+b{x}^{3}+cx+d$$ has zero gradient at the point $$(0,1)$$ and also touches the x-axis at the point $$(-1,0)$$ then