Find the equation of the locus of P, if A = (2, 3), B = (2, –3) and

1.	Find the equation of the locus of P, if A = (2, 3), B = (2, –3) and PA + PB = 8.
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

Find the equation of the locus of P, if A = (2, 3), B = (2, –3) and PA + PB = 8.

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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