The point(s) at each of which the tangents to the curve ;$\displaystyle y = x^3 - 3x^2 - 7x + 6$ cut off on the positive semi axis $OX$ a line segment half that on the negative semi axis $OY$, then the co-ordinates of the point(s) is/are give by:

$y={ x }^{ 3 }-3{ x }^{ 2 }-7x+6\\ \dfrac { dy }{ dx } =3{ x }^{ 2 }-6x-7$

Let coordinate of X be $\left( a,0 \right)$ then coordinate of Y is $\left( 0,-2a \right)$

Therefore slope of XY is $\cfrac { 0+2a }{ a-0 } =2$

Equating slope $3{ x }^{ 2 }-6x-7=2\\ 3{ x }^{ 2 }-6x-9=0$

Solving we get

$x=-1,y=9\quad \\ x=3,y=-15$

Hence, option 'B' is correct.