Tangents and it...

Equation of the tangent to the curve  $$y(x-2)(x-3)-x+7=0$$ at the point where
the curve cuts x-axis is

P(1, 1) is a point on the parabola  $$y=x^{2}$$ whose vertex is A. The point on the curve at which the tangent drawn is parallel to the chord  $$\overline{AP}$$   is

Equation of the tangent to the parabola $$y^{2}=4x+5$$ which is parallel to the line $$y= 2x + 7$$ is

For the parabola $$y^{2}=8x$$, tangent and normal are drawn at $$P(2, 4)$$ which meet the axis of the parabola in $$A$$ and $$B$$, then the length of the diameter of the circle through $$A, P, B$$ is

If the curves $$y=x^{2}-1,\ y=8x-x^{2}-9$$  touch each other at (2, 3) then equation of the common tangent is

The point on the hyperbola $$y = \dfrac {x - 1}{x + 1}$$ at which the tangents are parallel to $$y = 2x + 1$$ are

Assertion(A): The tangent to the curve $$y=x^{3}-x^{2}-x+2$$ at (1, 1) is parallel to the x axis.
Reason(R): The slope of the tangent to the above curve at (1, 1) is zero.

Assertion A: The curves $$x^{2}=y,\ x^{2}=-y$$  touch each other at (0, 0).
Reason R: The slopes of the tangents at (0, 0) for both the curves are equal.

Observe the following statements for the curve $$y=2.e^{\frac{-x}{3}}$$
I : The slope of the tangent to the curve where it meets y-axis is $$\displaystyle \frac{-2}{3}$$
II:The equation of normal to the curve where it meets y-axis is $$3x+2y+4=0$$.
Which of the above statement is correct

Match the points on the curve  $$2y^{2}=x+1$$ with the slope of normals at those points and choose