Question

If graph of curve $f ( x )=\{\sin B -\sin ( B + C )\} x ^2+\{\sin A -\sin ( A + B )\} x +\{\sin C -\sin ( C + A )\}$ touches $x$ axis at a point lying between roots of quadratic equation $( K -5) x ^2-2 kx + k -4=0$, then find least integral value of $k$. (where $A , B , C$ are angles of a triangle.)

Answer 1

$\Theta f(x)=(\sin B-\sin A ) x^2+(\sin A -\sin C ) x +(\sin C -\sin A )$
Clearly $x=1$ is a root of equation.
$\because $ Curve touches $x$ axis.
$\therefore $ both roots are equal to 1 .
$\because 1$ lies between roots.
$\therefore ( k -5) f (1)<0$
$\Rightarrow ( k -5)( k -5-2 k + k -4)<0$
$\Rightarrow ( k >5)$, least value $=6$