Look
here for a plot of the graph defined by your parametric equations (with
a=1). You'll notice it has a sharp point at
(0,1) where
t=0 -- this is similar to what happens to the graph
y=∣x∣ at
x=0. It's no coincidence that
dxdy is undefined at sharp points, because the notion of a "gradient" doesn't make sense there.
Going back to first principles, the gradient of a function
f at a point
x is given by the limit of
hf(x+h)−f(x) as
h→0 (if this limit exists). In the example
y=∣x∣ we have:
dxdy=h→0limh∣x+h∣−∣x∣=h→0limh(∣x+h∣+∣x∣)(∣x+h∣−∣x∣)(∣x+h∣+∣x∣)=h→0limh(∣x+h∣+∣x∣)(x+h)2−x2=h→0lim∣x+h∣+∣x∣2x−hThis is all well and good when
x=0; taking the limit simply gives
∣x∣x which is as we expect: 1 when x is positive and -1 when it's negative. But when x=0, we have to take the limit of
∣h∣−h as
h→0, which changes value depending on whether h is coming from above or below, meaning the gradient is undefined.
Another way of thinking about it is deciding where the tangent would go. If you think of the tangent to a curve at a point as being the limit of a cord on the curve as the two points of intersection of the cord and the curve tend to the point in question, you could essentially generate a "tangent" of any gradient between -1 and 1, simply by choosing cords which intersect the curve at different points and slide towards the origin at different speeds.
If any of the above makes no sense to you, don't worry too much about it