Application of ...

The points on the curve $$\displaystyle y^{2}=4a\left ( x+a\sin \frac{x}{a} \right )$$ at which the tangent is parallel to x axis lie on -

A stone is dropped into a quiet lake and waves move in a circle at a speed of $$3.5 cm/sec$$. At the instant when the radius of the circular wave is $$7.5 cm$$. The enclosed area increases as fastly as.

The normal to the curve $$\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}$$ is perpendicular to $$x$$ axis at the point

If equation of normal at a point $$\displaystyle \left ( m^{2},-m^{3} \right )$$ on the curve $$\displaystyle x^{3}-y^{2}=0\: \: is\: \: y=3mx-4m^{3}$$ then $$\displaystyle m^{2}$$ equals

The slope of the tangents to the curve $$y = (x + 1) (x - 3)$$ at the points where it crosses x - axis are

The points on the curve $$\displaystyle 9y^2=x^{3}$$ where the normal to the curve makes equal intercepts with coordinates axes is :

The coordinates of the points on the curve $$\displaystyle x=a\left ( \theta +\sin \theta  \right ),y=a\left ( 1-\cos \theta  \right )$$ where tangent is inclined an angle $$\displaystyle \dfrac{\pi }4$$ to the $$x-$$axis are -

If the curve $$y^2=ax^3-6x^2+b$$ passes through $$(0,\,1)$$ and has its tangent parallel to y-axis at $$x=2$$, then

Let tangent at a point P on the curve $$\displaystyle { x }^{ 2m }={ y }^{ \tfrac { n }{ 2 }  }={ a }^{ \tfrac { 4m+n }{ 2 }  }$$ meets the x-axis and y-axis at A and B respectively, If AP:PB is $$\displaystyle \frac { n }{ \lambda m } $$, where P lies between A and B, then find the value of $$\displaystyle \lambda $$

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