Find the number of values of 'a' for which the curves $y= ax^2 + ax + \frac{1}{24}$ and $x = ay^2 + ay + \frac{1}{24}$ touch each other.
MY ATTEMPT:
Since both the curves are the inverse of each other, they must be images about the line $y = x$. Therefore, for them to touch each other, they must touch $y=x$
Thus I solved the line with one of the curves and then applied the condition for equal roots ($D=0$)
So I got two values of a as $\frac23$ and $\frac32$
However the correct answer is 4 values of a. So I checked out the graph of the curves (by substituting 'a' with the other answers) and discovered that there is also a possibility of them touching on the line $y=x$ without touching the line itself as shown here
How do I find the other two values of 'a' in this scenario?