Show that there is no cubic curve for which the tangent lines at two distinct points coincide.

In how many points two distinct lines can intersect?

Statement 1: The point of intersection of the tangents at three distinct points A,B, and C on the parabola y^(2)=4x can be collinear. Statement 2: If a line L does not intersect the parabola y^(2)=4x, then from every point of the line,two tangents can be drawn to the parabola.

A line which intersects a circle at two distinct point is called a ______ of the circle.

(ii) No two line segments with a common end points are coincident.

Find the points on the curve x^3-2x^2-2x at which the tangent lines are parallel to the line y=2x-3 .

Find the points on the curve y=x^3-2x^2-x at which the tangent lines are parallel to the line y=3x-2 .

Prove that the curve y=e^(|x|) cannot have a unique tangent line at the point x = 0. Find the angle between the one-sided tangents to the curve at the point x = 0.

Prove that , "If a line parallel to a side of a triangle intersects the remaining sides in two distinct points then the line divides the sides in the same proportion".

At what points on the curve x^(3)+y^(3)=6xy is the tangent line horizontal?