Question

Two points A(1,0)&B(-1,0) with a variable point P(x,y) satisfy the relation AP-BP =1 Show that 12x^2-4y^2=3

Answer 1

12x²  - 4y² = 3 if Two points A(1,0)&B(-1,0) with a variable point P(x,y) satisfy the relation AP-BP =1

Step-by-step explanation:

P = (x , y)

A = ( 1, 0)

B = ( - 1, 0)

AP  = √(x - 1)² + (y - 0)²

AP = √(x - 1)² + y²

BP  = √(x - (-1))² + (y - 0)²

BP = √(x + 1)² + y²

AP - BP  = 1

=> √((x - 1)² + y²)  -   √((x + 1)² + y²)   =  1

=> √((x - 1)² + y²) = 1 + √((x + 1)² + y²)

Squaring both sides

=> (x - 1)² + y²  =  1 + (x + 1)² + y²  + 2√((x + 1)² + y²)

=> -4x - 1  = 2√((x + 1)² + y²)

Squaring both sides

=> 16x² + 1 + 8x  = 4((x + 1)² + y²)

=>  16x² + 1 + 8x  = 4(x² + 1 + 2x + y²)

=>  16x² + 1 + 8x  = 4x² + 4 + 8x + 4y²

=> 12x²  - 4y² = 3

QED

Proved

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Answer 2

Answer:

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