Tangents and it...

Line $$y=x$$ and curve $$y=x^2+bx+c$$ touches at $$(1, 1)$$ then __________.

Equation of the normal of curve $$y=x^3-2x+4$$ at point $$(1, 3)$$ is __________.

The equation to the tangent to $${\left( {{\dfrac xa}} \right)^n} + {\left( {{\dfrac yb}} \right)^n} = 2$$ at $$\left( {a,b} \right)$$

Find the equations of the tangent and the normal to the curve $$x^2 +y^2=5$$ where the tangent is parallel to the lines $$x-y =0$$ respectively.

The point on the curve $$y^{2}=x$$ where tangent makes $$45^{o}$$ angle with $$x-$$axis ?

Equation of the normal to the curve $$y = -\sqrt{x} + 2$$ at the point of its interaction with the curve $$y = x$$ is

The perpendicular distance from origin to the normal at any point to the curve $$x = a\left( {\cos \theta  + \theta \sin \theta } \right)$$. $$y = a\left( {\sin \theta  - \theta \cos \theta } \right)$$

The slope of the tangent and normal to $$y={ x }^{ 2 }-3x+5$$ at $$(2,3)$$ are ______ and _____ respectively.

Find the angle between tangent of the curve $$y = (x + 1) (x - 3)$$ at the point where it cuts the axis of $$x$$.

If the tangent at the P of the curve $${y^2} = {x^3}$$ intersect the curve again at Q and the straight lines OP, OQ make angles $$\alpha ,\beta $$ with the x-axis, where O is the origin. then, $${{\tan \alpha } / {\tan \beta }}$$ has the value equal to