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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?

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  • $\begingroup$ 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) $\endgroup$
    – imranfat
    Apr 26 '16 at 20:39
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The tangent line at a point is not defined as the unique line that intersects the curve at that point alone; for example, a tangent line for a trig function can intersect the graph in another point. It is true that the tangent line intersects the graph at exactly one point only when the function is uniformly and strictly concave or convex; in this case, it is concave right, so each tangent line, like the example you noted, intersects the curve in exactly one point. However, it is the reverse statement that is false for your example, the uniqueness of the tangent line in this respect. For example, the tangent line to a line at any given point on the line is clearly the line itself, even though any other line through the point intersects the graph only at the point in question. These are, however, not tangent lines. Also, in this case, the actual tangent line intersects the curve (line) in (infinitely many) other points, for the line is not uniformly strictly concave or convex. The definition of the tangent line is the line that best follows the curve near the point in question; it doesn't actually have to do with other intersections between the line and the curve.

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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 $ax^2+2hxy+by^2+2gx+2fy+c=0$.

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