Question

find the equation of the locus of p if A(2,3) B(,2,-3)and PA+PB=8​

Answer 1

Answer:

Given A = (2, 3)

B = (2, –3)  

Let P(x, y) be any point on the locus.  

Given geometric condition is  

PA + PB = 8 ⇒ PA = 8 – PB  

⇒ PA² = (8 – PB)²

⇒ PA² = 64 + PB² – 16PB

⇒ (x – 2)²+(y – 3)²= 64 + (x – 2)²+ (y + 3)²- 16√((x - 2)²+ (y + 3)²)

⇒ y² – 6y + 9 – y² – 6y – 9 – 64 = – 16√(x² - 4x + 4 + y² + 6y + 9)

⇒ (–12y – 64) = –16√(x² + y² - 4x + 6y + 13)

⇒ (3y + 16) = 4√(x² + y² - 4x + 6y + 13)

 S.O.B.S

⇒ (3y + 16)² = 16 (x² + y² – 4x + 6y + 13)

⇒ 9y² + 256 + 96y = 16x² + 16y² – 64x + 96y + 208  

⇒ 16x² + 7y² – 64x – 48 = 0

∴ The locus of P is 16x² + 7y² – 64x – 48 = 0

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