If the lines $k x+2 y-4=0$ and $5 x-2 y-4=0$ are conjugate with respect to the circle $x^2+y^2-2 x-2 y+1=0$, then $k$ is equal to

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Aicharya · Accepted Answer

The lines $k x+2 y-4=0$ and $5 x-2 y-4=0$ are conjugate with respect to the circle $x^2+y^2-2 x-2 y+1=0$ Then, necessary condition $\begin{aligned} & r^2\left(a_1 a_2+b_1 b_2ight)=\left(a_1 g+b_1 f-c_1ight) \ & \left(a_2 g+b_2 f-c_2ight) \end{aligned}$ From the above equation, $\begin{aligned} & a_1=k, \quad b_1=2 \quad c_1=-4 \ & a_2=5, \quad b_2=-2 \quad c_2=-4 \ & g=-1, \quad f=-1 \quad c=1 \ & r^2=g^2+f^2-\mathrm{c}=1+1-1=1 \end{aligned}$ Now, by using equation (i) $\begin{aligned} & 1(5 k-1)=(-k-2+1)(-5+2+4) \ & 5 k-4=-k+2 \ & 6 k=6 \ & \Rightarrow \quad k=1 \end{aligned}$

If the lines $k x+2 y-4=0$ and $5 x-2 y-4=0$ are conjugate with respect to the circle $x^2+y^2-2 x-2 y+1=0$,…

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