Question

Find $\lambda$, if the distance between $(\lambda ,1)$ and $(1,-\lambda )$ is $2\sqrt{5}$ units.

Answer 1

${(\lambda -1)}^2+{(1-\lambda )}^2={\left(2\sqrt{5}\right)}^2$

$\Rightarrow {\lambda }^2+1-2\lambda +{\lambda }^2+1-2\lambda =20$

$\Rightarrow 2{\lambda }^2-4\lambda +2-20=0$

$\Rightarrow 2{\lambda }^2-4\lambda -18=0$

$\Rightarrow {\lambda }^2-2\lambda -9=0$

$\lambda =\frac{2\pm \sqrt{4+4\times 9}}{2}=\frac{2\pm \sqrt{4+36}}{2}$

$=\frac{2\pm \sqrt{40}}{2}=\frac{2\pm 2\sqrt{10}}{2}$

$=1\pm \sqrt{10}$