Tangents and it...

Tangents are drawn from origin to the curve $$y=\sin{x}+\cos{x}$$. Then their points of contact lie on the curve

The slope of normal to the curve y= log (logx) at x = e is 

The inclination of the tangent at $$\theta = \dfrac {\pi}{3}$$ on the curve $$x = a(\theta + \sin \theta), y = a(1 + \cos \theta)$$ is

Let f be a real-valued differentiable function on R (the set of all real numbers) such that $$f(1)=1$$. If the y-intercept of the tangent at any point P(x, y) on the curve $$y=f(x)$$ is equal to the cube of the abscissa of P, then the value of $$f(-3)$$ is equal to?

The greatest slope among the lines represented by the equation $$4x^2 - y^2 + 2y - 1 = 0 $$ is - 

If the tangent at $$(x_{1}, y_{1})$$ to the curve $$x^{3}+y^{3}=a^{3}$$ meets the curve again at $$(x_{2}, y_{2})$$ then

The ordinate of all points on the curve $$y=\dfrac{1}{2\sin^{2}x+3\cos^{2}x}$$  where the tangent is horizontal, is

The curve given by $$x + y = {e^{xy}}$$ has an tangents parallel to the y-axis at the point

A curve C has the property that if the tangent drawn at any point 'P' on C meets the coordinate axes at A and B, and P is midpoint of AB. If the curve passes through the point $$(1, 1)$$ then the equation of the curve is?

Number of possible tangents to the curve $$y = \cos \left( {x + y} \right), - 3\pi  \leqslant x \leqslant 3\pi $$, that are parallel to the line $$x + 2y = 0$$, is