Tangents and it...

The slope of the tangent to the locus $$y=\cos^{-1}\left ( \cos x \right )$$ at $$x=\displaystyle \frac{\pi }{4}$$ is

$$P(2, 2)$$ and $$Q\left ( \displaystyle \frac{1}{2}, -1 \right )$$ are two points on the parabola $$y^{2}=2x$$. The coordinates of the point $$R$$ on the parabola, where tangent to the curve is parallel to the chord $$PQ$$ is

The number of tangents to the curve $$y^{2}-2x^{3}-4y+8=0$$ that pass through $$(1, 0)$$ is

The curve given by $$x+y=e^{xy}$$ has a tangent as the $$y$$-axis at the point

Let $$\displaystyle f(x)=e\:^{x}\sin x$$ be the equation of a curve. If at $$\displaystyle x=a,0\leq a\leq 2\pi$$, the slope of the tangent is the maximum then the value of $$a$$ is 

The equation of tangent to the curve $$y=e^{-\left | x \right |}$$ at the point where the curve cuts the line $$x=1$$ is

The equation of the curve is given by $$x=e^{t}\sin t$$, $$y=e^{t}\cos t$$. The inclination of the tangent to the curve at the point $$t=\displaystyle \frac{\pi }{4}$$ is

If the curve $$y=ax^3+bx^2+cx$$ is inclined at $$45^0$$ to the x-axis at $$(0,0)$$ but it touches the x-axis at $$(1,0)$$ then the value of $$a,b$$ and $$c$$ are given by

The point on the curve $$\sqrt{x}+\sqrt{y}=2a^{2}$$, where the tangent is equally inclined to the axes, is

The angle between two tangents to the ellipse $$\displaystyle \frac{x^{2}}{16}+\frac{y^{2}}{9}=1$$ at the points where the line $$y=1$$ cuts the curve is