Question

What loci are represented by the equations :
$(y-mx-c{)}^2+(y-m^{\prime }x-c^{\prime }{)}^2=0.$

Answer 1

Given equation is ${(y-mx-c)}^2+{(y-m^{\prime }x-c^{\prime })}^2=0$

Both the terms in the given equation are prefect square of two straight lines.

Their sum can only be zero if they are simultaneously zero.

So, the equation represents point of intersection of

$y-mx-c=\mathrm{0.......}\left(i\right)$ and $y-m^{\prime }x-c^{\prime }=\mathrm{0......}\left(ii\right)$

On solving $\left(i\right)$ and $\left(ii\right)$

$\Rightarrow x=\frac{c^{\prime }-c}{m-m^{\prime }},y=\frac{m{c}^{\prime }-c{m}^{\prime }}{m-m^{\prime }}$

So, loci is the point $(\frac{c^{\prime }-c}{m-m^{\prime }},\frac{m{c}^{\prime }-c{m}^{\prime }}{m-m^{\prime }})$