Tangents and it...

The curve $$y+e^{xy}+x= 0$$ has a tangent parellel to y-axis at a point

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

The normal to the curve $$x=a\left ( 1-\cos \theta  \right )$$, $$y=a\sin \theta $$ at $$\theta $$ always passes through the fixed point

If the normal to the curve $$\displaystyle y= f\left ( x \right )$$ at the point $$\displaystyle \left ( 3, 4 \right )$$ makes an angle $$\displaystyle \frac{3\pi}{4}$$ with the positive x-axis then $$\displaystyle f'\left ( 3 \right )$$ is equal to

If the tangent at P  on the curve $$\displaystyle x^{m}y^{n}=a^{m+n}$$ meets the co-ordinates axes at A and B, then $$AP: PB= $$

Find the equation of the tangent to the curve at any point $$(X, Y)$$.

For the equation $$\displaystyle x^{2/3}+y^{2/3}=a^{2/3}$$, find the equation of tangent at the point $$\displaystyle x=a\sin ^{3}\theta, y=a\cos ^{3} \theta$$.

Find the condition that the line $$\displaystyle Ax+By= 1$$ may be a normal to the curve $$\displaystyle a^{n-1}y=x^{n}.$$

What are the tangent and normal to the curve $$x=\displaystyle \frac{2at^{2}}{1+t^{2}}$$, $$ y= \displaystyle \frac{2at^{3}}{a+t^{2}}$$ at the point for which $$\displaystyle t=\frac{1}{2}$$

Normal to the curve $$y=\displaystyle x^{3}-2x^{2}+4$$ at the point where $$x=2$$