Application of ...

If the circle $$x^2+y^2+2gx+2fy+c=0$$ is touched by $$y=x$$ at P such that OP = $$6\sqrt{2}$$
then the value of c is

The abscissas of points $$P$$ and $$Q$$ on the curve $$y=e^x+e^{-x}$$ such that tangents at $$P$$ and $$Q$$ make $$60^{\circ}$$ with the $$x$$-axis are

The graphs $$y=2x^3-4x+2$$ and $$y=x^3+2x-1$$ intersect at exactly 3 distinct points. The slope of the line passing through two of these points

Let $$S$$ be a square with sides of length $$x$$. If we approximate the change in size of the area of $$S$$ by $$\displaystyle h.\frac{dA}{dx}|_{x=x_0}$$, when the sides are changed from $$x_0$$ to $$x_o+h$$, then the absolute value of the error in our approximation, is

The number of points on the curve $$x^{3/2}+y^{3/2}=a^{3/2}$$, where the tangents are equally inclined to the axes, is

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

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 angle made by the tangent of the curve $$\displaystyle x = a(t + \sin t \cos t); y = a (1 + \sin t)^2$$ with the x-axis at any point on it is

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

Let $$\displaystyle \:f : R\rightarrow R$$ be a function such that $$\displaystyle \:f \left ( x \right )= ax+3\sin x+4\cos x.$$ Then $$\displaystyle \:f \left ( x \right )$$ is invertible if