Tangents and it...

The slope of the tangent to the curve $${r^2} = {a^2}\cos 2\theta$$, where $$x = r\cos \theta ,y = r\sin \theta $$, at the point $$\theta=\frac{\pi}{6}$$ is

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

If the tangent at any point on the curve $$x^{4} +y^{4}=a^{4}$$ cuts off intercepts $$p$$ and $$q$$ on the coordinate axes the value of $$p^{-4/3}+q^{-4/3}$$ is

The abscissa of the point on the curve $$\sqrt {xy}  = a + x$$ , the tangent at which cuts off equal intercepts from the co-ordinate axes is (a >0)

if $$m$$ is the slope of a tangent to the curve $$e^{y}=1+x^{2}$$, then  $$m$$ belongs to the interval

Find the slope of tangent of the curve$$x = a\,{\sin ^3}t,y = b\,\,{\cos ^3}t$$ at $$t = \frac{\pi }{2}$$

A tangent drawn to the curve $$y = f\left( x \right)$$ at $$P\left( {x,y} \right)$$
cuts the x and y axes at A and B, respectively, such that $$AP:PB = 1:3$$. If $$f\left( 1 \right) = 1$$ then the curve passes through $$\left( {k,\frac{1}{8}} \right)$$ where $$k$$ is

Equation of tangent to the circle $$x^{2}+ y^{2}-6x+4y-12=0$$ which are parallel to the line $$4x+3y+5=0$$ is ?

The point on the curve $$y = b e^{\dfrac {-x}{a}}$$ at which the tangent drawn is $$\dfrac {x}{a} + \dfrac {y}{b} = 1$$ is

If $$V$$ is the set of points on the curve $$y^{3} - 3xy +2 = 0$$ where the tangent is vertical then $$V =$$.