If AP2−BP2=18, where A ≡ (1, 2, –3) and B ≡ (3, –2, 1). Then the locus of point P is
We want to find the locus of P. That is, the set of Ps satisfying the given condition. For that, let the point P be (x, y, z). Using distance formula, we get
AP2=(x−1)2+(y−2)2+(z+3)2
BP2=(x−3)2+(y+2)2+(z−1)2
Given that AP2−BP2=18
⇒(x−1)2+(y−2)2+(z+3)2−{(x−3)2+(y+2)2+(z−−1)2}=18
⇒x2+1−2x+y2+4−4y+z2+9+6z−{x2+9−6x+y2+4+4y+z2+1−2z}=18
⇒x2+y2+z2−2x−4y+6z+14−x2−y2−z2+6x−4y+2z−14=28
⇒ 4x–8y+8z=18
⇒ 2x–4y+4z=9
Locus of the point P is 2x – 4y + 4z = 9