Tangents and it...

The normal curve $$xy = 4$$ at the point $$(1, 4)$$ meets the curve again at

The tangent to the curve $$y={e}^{x}$$ drawn at the point $$\left(c,{e}^{c}\right)$$ intersects the line joining the points $$\left(c-1,{e}^{c-1}\right)$$ and $$\left(c+1,{e}^{c+1}\right)$$

The point (s) on the curve $$\displaystyle y^{3}+3x^{2}= 12y,$$ where the tangent is vertical (i.e., parallel to the y-axis),  is / true

Find the equations of tangents to the curve y=x$$^{4}$$ which are drawn from the point $$(2,0)$$

A figure is bounded by the curve $$\displaystyle y=x^{2}+1,$$ the axes of co-ordinates and the line x=1. Determine the co-ordinates of a point P at which a tangent be drawn to the curve so as to cut off a trapezium of greatest area from the figure.

At which point the tangent to $$\displaystyle x^{3}= ay^{2}$$ at $$\displaystyle \left ( 4am^{2},8am^{3} \right )$$ cuts the curve again.

Let the equation of a curve be $$x=a\left ( \theta +\sin \theta  \right )$$, $$y=a\left ( 1-\cos \theta  \right )$$. If $$\theta $$ changes at a constant rate $$k$$ then the rate of change of slope of the tangent to the curve at $$\displaystyle \theta =\frac{\pi }{2}$$ is

The positive value of $$k$$ for which $$ke^x-x=0$$ has only one real solution is 

Find the co-ordinates of the point (s) on the curve $$\displaystyle y= \frac{x^{2}-1}{x^{2}+1}, x> 0$$ such that tangent at these point (s)have the greatest slope.

The point of intersection of the tangents drawn to the curve $$\displaystyle x^{2}y= 1-y$$ at the point where it is intersected by the curve xy$$=1-y,$$ is given by