Question

Determine the points on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, -4)

Answer 1

Let P(0, 0, z) be the point equidistant from Q(1, 5, 7) and R(5, 1, -4)

So,

$(PQ{)}^2=(PR{)}^2$

$\Rightarrow (0-1{)}^2+(0-5{)}^2+(z-7{)}^2=(0-5{)}^2+(0-1{)}^2+(z+4{)}^2$

$\Rightarrow 1+25+(z-7{)}^2=25+1+(z+4{)}^2$

$\Rightarrow 26+z^2+49-14z=26+z^2+8z+16$

$\Rightarrow -14z-8z=16-49$

$\Rightarrow -22z=-33$

$\Rightarrow z=\frac{-33}{-22}$

$\Rightarrow z=\frac{3}{2}$

$\text{Required point}=(0,0,\frac{3}{2})$