Tangents and it...

The coordinates of the points(s) at which the tangents to the curve $$\displaystyle y=x^{3}-3x^{2}-7x+6$$ cut the positive semi axis OX a line segment half that on the negative semi axis OY is/are given by

The tangent to the curve $$y=e^{x}$$ drawn at the point $$\left ( c,e^{c} \right )$$ intersects the line joining the points $$(c -1,e^{c-1})$$ and $$(c +1,e^{c+1}) $$

For the curve $${x}^{2}+4xy+8{y}^{2}=64$$ the tangents are parallel to the $$x$$-axis only at the points

The equation of the normal to the curve $$y(1+{x}^{2})=2-x$$ where the tangent crosses $$x$$-axis is 

If $$y = 4x - 5$$ is a tangent to the curve $$y^{2} = px^{3} + q$$ at $$(2, 3)$$, then

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

The sum of intercepts of the tangent to the curve $$\sqrt { x } +\sqrt { y } =\sqrt { a } $$ upon the coordinates axes is

Suppose that the equation $$f(x)={x}^{2}+bx+c=0$$ has two distinct real roots $$\alpha,\beta$$. The angle between the tangent to the curve $$y=f(x)$$ at the point $$\left( \cfrac { \alpha +\beta  }{ 2 } ,f\left( \cfrac { \alpha +\beta  }{ 2 }  \right)  \right) $$ and the positive direction of the x-axis is

If the slope of the curve $$y=\cfrac { ax }{ b-x } $$ at the point $$(1,1)$$ is $$2$$, then the values of $$a$$ and $$b$$ are respectively

The two curves $$x^{3} - 3xy^{2} + 2 = 0$$ and $$3x^{2} y - y^{3} = 2$$