A point \(P\) on the x-axis is at the same distance from \(A(2,\,-5)\) and \(B(-2,\,9)\). What are the coordinates of \(P\)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \((-7,\,0)\)
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.
\(PA^2=(x-2)^2+(0+5)^2=x^2-4x+4+25\)
\(PB^2=(x+2)^2+(0-9)^2=x^2+4x+4+81\)
Equal squares: \(x^2-4x+29=x^2+4x+85\)
The \(x^2\) terms cancel: \(-8x=56\), so \(x=-7\)
So \(P=(-7,\,0)\)
Check: \(PA^2=81+25=106\) and \(PB^2=25+81=106\)
