(a)

The equation of the parabola is .

The two lines that do not intersect are parallel  means the slopes are identical.

The slope of the tangent line is the same as he parabola at the tangent point.

Consider .

Apply derivative on each side with respect to .

.

The value of are distinct for all values.

Hence, it is impossible to have two distinct parallel tangent lines to a parabola.

Therefore, all pairs of tangent lines intersect.

(b) The equation of the parabola is at the points and .

Apply derivative on each side with respect to .

.

The derivative of the function at the point is .

The tangent line of the parabola at is .

.

The derivative of the function at the point is .

The tangent line of the parabola at is

Substitute .

.

Substitute in .

.

Therefore, the point of the intersection lines is at .

(a) All pairs of tangent lines intersect.

(b) The point of the intersection lines is at .