# Can a line parallel to axis of parabola also represent tangent at a point along with the one whose slope is found using calculus?

Consider a parabola with the equation $y^2=4x$ its axis is the x-axis and vertex is (0,0) and focus at (1,0). Consider any point on the parabola say (4,4). Now we define tangent at this point as a line that intersects the parabola at one and only one point which is (4,4). and no else. Hence by using derivative slope of tangent will be $1 \over 2$and its equation will be $x-2y+4=0$ But can the straight line parallel to axis of parabola with equation $y=4$ also be tangent at this point as it only cuts parabola at one point ? (Also this cannot be the case with the vertex (0,0) as then the line $y=0$ will represent normal at this point not tangent.) Can this somehow be linked with points at infinity or not?

• Not sure exactly was you are asking. But if I am right, you want to know if a line parallel to the parabola's axis of symmetry could be a tangent to the parabola at some point? That answer would be: No. (Unless you would consider infinity as a possible coordinate) Apr 26 '16 at 20:39

Consider the line $$x-2y+4=0$$ and the parabola $$y^2=4x$$. Now plug in $$y=\frac{x+4}{2}$$ from the line to the parabola $$y^2=4x$$. We get $$\left(\frac{x+4}{2}\right)^2=4x$$. On simplification we get $$x^2-8x+16=0$$ or $$(x-4)^2=0$$ or $$x=4,4$$ (repeated root), which gives the abscissa of the point of intersection of the given line and the quadratic curve $$y^2=4x$$. Thus we can say that the given line is a tangent line because it intersects the parabola at two points which are coincident points $$(4,4)$$ and $$(4,4)$$.
However if we consider the line $$y=0$$ or any line parallel to axis of the parabola; and the parabola $$y^2=4x$$ in a similar fashion we would get the equation $$x=0$$ giving the abscissa of the point of intersection of the given line and the parabola. Thus the point of intersection in this case is $$(0,0)$$ just simple intersection. Hence the line $$y=0$$ is not a tangent line.
We can test for any line to be a tangent or not in this way by plugging $$x$$ or $$y$$ from the line to any quadratic curve $$ax^2+2hxy+by^2+2gx+2fy+c=0$$.