I came across the following problem:

```
   For y=x³, tangent at A meets the curve again at B. 
   Gradient at B is k times the gradient at A. Then the 
   number of integral values of k is:

```

I proceeded as:-

Let A (a,y1) and B (b,y2) **for all real x**

The slope of tangent at B = (a³-b³)/(a-b)

And we know that slope of tangent at A, that is, 3a² = K times slope at B

So, process this information, I got the following quadratic equation:

```
      (3-K)(b/a)² - K(b/a) - K = 0

```

Since x is real, the discriminant of the above equation has to be greater than or equal to zero. Processing it I got:

```
        K belongs to the interval [0,4]

```

So the answer came out to be 5, since we need the integral values. But in the answer key, it is given that k cannot be zero, thus making the answer 3.

My question is why can't K be zero? Am I missing something?

*Please note that a similar question has been asked by someone, but the answer is not at all satisfactory. Thus, please refrain from providing any links to that answer*