Question

A(1,2) B(2,-3) C(-2,3) are 3 points A point moves such that PA²+PB²=2PC²
show that the equation to the locus of P is 7x-7y+4=0​

Answer 1

Let point P be (x,y)

Given A(1,2),B(2,−3),C(−2,3) and

PA  

2

+PB  

2

=2PC  

2

 

⟹(  

(x−1)  

2

+(y−2)  

2

 

​  

)  

2

+(  

(x−2)  

2

+(y+3)  

2

 

​  

)  

2

=2(  

(x+2)  

2

+(y−3)  

2

 

​  

)  

2

 

⟹x  

2

+1−2x+y  

2

+4−4y+x  

2

+4−4x+y  

2

+9+6y=2(x  

2

+4+4x+y  

2

+9−6y)

⟹2x  

2

+2y  

2

−6x+2y+18=2x  

2

+2y  

2

+8x−12y+26

⟹14x−14y+8=0

⟹7x−7y+4=0

Therefore, the locus of point P is 7x−7y+4=0.

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