UPSC NDA Math Model Paper 6

Given, let P(x,y) be any point. A = (1, 2) and B = (-2,3).
A point P (x, y) moves such that its distances from A = (1, 2) and B = (-2, 3) are equal.
⇒ PA = PB
⇒(PA)2=(PB)2
By Distance formula,
(x−1)2+(y−2)2=(x+2)2+(y−3)2
⇒x2−2x+1+y2−4y+4 =x2+4x+4+y2−6y+9
⇒6x−2y+8=0
⇒3x−y+4=0 which represent the equation of line.
Hence, the locus of P is a straight line