The point \((x, y)\) is equidistant from the points \((2, 4)\) and \((1, 8)\). Find the relation between x and y.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(2x-8y=-45\)
A point equidistant from two points has equal distances to both, so the two squared distances (by the distance formula) are equal. The \(x^{2}\) and \(y^{2}\) terms appear on both sides and cancel, so a linear relation is left.
Squared distance to (2, 4) equals squared distance to (1, 8): \((x-2)^{2}+(y-4)^{2}=(x-1)^{2}+(y-8)^{2}\)
Expanding both sides: \(x^{2}-4x+4+y^{2}-8y+16=x^{2}-2x+1+y^{2}-16y+64\)
After the squares cancel: \(-4x-8y+20=-2x-16y+65\)
So \(-2x+8y=45\), that is, \(2x-8y=-45\)