Question

In the diagram on a lunar eilpse, if the positions od sun,Earth and moon are shown by $(-4,6),(k,-2)$and$(5,-6)$ respectively, then find the value of k.

Answer 1

As the sun, earth and moon lie on the same axis. Thus the points $(-4,6),(k,-2)$ and $(5,-6)$ are collinear.

Therefore,

$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

$\left|\begin{array}{ccc}-4& 6& 1\\ k& -2& 1\\ 5& -6& 1\end{array}\right|=0$

$-4(-2-(-6))-6(k-5)+1(-6k-(-10))=0$

$-16-6k+30-6k+10=0$

$24-12k=0$

$\Rightarrow k=\frac{24}{12}=2$

Hence the value of $k$ is $2$.