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 $(K-5)x^2-2kx+k-4=0$, then find least integral value of $k$. (where $A,B,C$ are angles of a triangle.)

Answer 1

Answer: 6

$\Theta f(x)=(\sin B-\sin A)x^2+(\sin A-\sin C)x+(\sin C-\sin A)$
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.
$\therefore (k-5)f(1)<0$
$\Rightarrow (k-5)(k-5-2k+k-4)<0$
$\Rightarrow (k>5)$, least value $=6$