Tangents and it...

Find the case of tangents & normal to the curve $$y=x^{4}-6x^{3}+13x^{2}-10x+5$$ at $$(1, 3)$$

Two lines drawn through the point $$A ( 4,0 )$$  divide the area bounded by the curve $$y = \sqrt { 2 } \sin ( \pi x / 4 )$$  and  $$x$$ - axis between the lines $$x = 2$$  and   $$x = 4$$  into three equal parts. Sum of the slopes of the drawn lines is:

The tangent to the curve $$2a^2y=x^3-3ax^2$$ is parallel to the x-axis at the points

The point on the curve $${x}^{2}+{y}^{2}-2x-3=0$$ at which the tangent in parallel to x-axis is 

Which of the following lines, is a normal to the parabola $${y}^{2}=16x$$?

The line $$3x-4y=0$$

The equation of the normal to the curve $$y^4=ax^3$$ at (a , a) is 

If $$x-2y+k=0$$ is a common tangent to $$\displaystyle{ y }^{ 2 }=4x\quad \& \frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { 3 } } =1\left( a>\sqrt { 3 }  \right)  $$, then the value of a, k and other common tangent are given by

The normal to the curve, $${{\text{x}}^{\text{2}}}{\text{ + 2xy - 3}}{{\text{y}}^{\text{2}}}{\text{ = 0,}}\;{\text{at}}\;\left( {{\text{1,1}}} \right){\text{:}}$$

The equation of one of the tangents to the curve $$y=\cos(x+y),-2\pi 
\le x \le 2\pi$$; that is parallel to the line $$x+ 2y = 0$$ , is