A point \(P\) on the x-axis is at the same distance from \(A(7,\,2)\) and \(B(-1,\,6)\). What are the coordinates of \(P\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \((1,\,0)\)
A point on the x-axis has y-coordinate 0, so let \(P=(x,\,0)\). \(PA=PB\) means the squares of the distances are equal, which removes the square roots of the distance formula.
\(PA^2=(x-7)^2+(0-2)^2=x^2-14x+49+4\)
\(PB^2=(x+1)^2+(0-6)^2=x^2+2x+1+36\)
Equal squares: \(x^2-14x+53=x^2+2x+37\)
The \(x^2\) terms cancel: \(-16x=-16\), so \(x=1\)
So \(P=(1,\,0)\)
Check: \(PA^2=36+4=40\) and \(PB^2=4+36=40\)
