Tangents and it...

A and B are points $$(-2,0)$$ and $$(1,3)$$ on the curve $$\displaystyle y=4-x^{2}$$. If the tangent at P on the curve be parallel to chord AB, then co-ordinates of point P are 

The parametric equations of a curve are given by $$\displaystyle x= \sec ^{2}t, y= \cot t.$$ Tangent at $$\displaystyle P \,t=\dfrac { \pi }4$$ meets the curve again at Q; then $$\displaystyle PQ=? $$

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

The curve $$\displaystyle y-e^{xy}+x=0$$ has a vertical tangent at the point 

If $$\displaystyle x\cos \alpha +y\sin \alpha =p$$ touches $$\displaystyle x^{2}+a^{2}y^{2}=a^{2},$$ then

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

The coordinates of the point $$M(x, y)$$ on $$\displaystyle y= e^{-\left | x \right |}$$ so that the area formed by the coordinates axes and the tangent at $$M$$ is greatest, are

The value of m for which the area of the triangle included between the axes and any tangent to the $$\displaystyle x^{m}y= b^{m}$$ curve is constant, is

If the tangent at any point on the curve $$\displaystyle x^{4}+y^{4}= a^{4}$$ cuts off intercepts p and q on the coordinate axes, the value of $$\displaystyle p^{-\tfrac 43}+q^{-\tfrac 43}$$ is

The equation of the tangent to the curve $$\displaystyle y= \left ( 2x-1 \right )e^{2\left ( 1-x \right )}$$ at the point of its maximum is