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Question

Find the locus of the point P if AP2BP2=18, where A ≡ (1, 2, –3) and B ≡ (3, –2, 1)


A

2x + 3y + 4 = 9

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B

2x + 4y + 4z = 9

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C

2x – 3y + 4z = 9

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D

2x – 4y + 4z = 9

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Solution

The correct option is D

2x – 4y + 4z = 9


We want to find the locus of P. That is, set of Ps satisfying the given condition. For that, let the point P be (x, y, z). Using distance formula, we get

AP2=(x1)2+(y2)2+(z+3)2

and

BP2=(x3)2+(y+2)2+(z1)2

Given that AP2BP2=18

(x1)2+(y2)2+(z+3)2{(x3)2+(y+2)2+(z1)2}=18

x2+12x+y2+44y+z2+9+6z{x2+96x+y2+4+4y+z2+12z}

x2+y2+z22x4y+6z+14x2y2z2+6x4y+2z14=28

4x8y+8z=18

2x4y+4z=9

Locus of the point P is 2x – 4y + 4z = 9


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