Question

Find a point which is equidistant from the points $A\left(-5,4\right)andB(-1,6).$How many such points are there ?

Answer 1

Let A(x, y) be the point which is equidistant from the points P(−5, 4) and Q(−1, 6).
Then AP = AQ.
$$\begin{gathered} \Rightarrow A{P}^2=A{Q}^2 \\ \Rightarrow {\left[x-\left(-5\right)\right]}^2+{\left[y-4\right]}^2={\left[x-\left(-1\right)\right]}^2+{\left[y-6\right]}^2 \\ \Rightarrow x^2+25+10x+y^2+16-8y=x^2+1+2x+y^2+36-12y \\ \Rightarrow 10x-2x-8y+12y+41-37=0 \\ \Rightarrow 8x+4y+4=0 \\ \Rightarrow 2x+y+1=0.....\left(i\right) \end{gathered}$$
Hence all the points satisfying the equation (i) are equidistant from the points P and Q.
There are infinite such points.