The sum of all integral values of $k(k
eq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=

Question

The sum of all integral values of $k(k 
eq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is

Professor PC Thomas Classes · Answer

(66) : Given, $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$\ $\Rightarrow \frac{2 x-4-x+1}{x^{2}-3 x+2}=\frac{2}{k}$\ $\Rightarrow \quad k(x-3)=2 x^{2}-6 x+4$\ $\Rightarrow \quad 2 x^{2}-(6+k) x+4+3 k=0$\ This equation has no real roots.\ $\Rightarrow D<0 \Rightarrow(k+6)^{2}-8(4+3 k)<0$\ $\Rightarrow \quad k^{2}+12 k+36-32-24 k<0$\ $\Rightarrow \quad k^{2}-12 k+4<0$\ $\Rightarrow \quad(k-6)^{2}<32$\ $\Rightarrow-4 \sqrt{2}<k-6<4 \sqrt{2}$\ $\Rightarrow 6-4 \sqrt{2}<k<6+4 \sqrt{2}$\ $\Rightarrow 0.3431<k<11.66$\ $	herefore \quad k$ can take values $1,2,3,4,5,6,7,8,9$, 10 and 11\ $	herefore \quad$ Sum of values of $k=\frac{11 	imes 12}{2}=66$\