Question

Find the locus of the point, the absolute value of difference of the distances of which from the points (2,2) and (0,0) is 2.identify the curve represented by the locus

Answer 1

The answer is given below :

Let us consider that the point is (x, y).

Then, the distance between the points (x, y) and (2, 2) is

= √{(x - 2)² + (y - 2)²} units

and the distance between the points (x, y) and (0, 0) is

= √{(x - 0)² + (y - 0)²} units

= √(x² + y²) units

By the given condition, the absolute value of difference of the above distances = 2

⇒ | √{(x - 2)² + (y - 2)²} - √(x² + y²) | = 2

⇒ √{(x - 2)² + (y - 2)²} - √(x² + y²) = 2,

⇒ √{(x - 2)² + (y - 2)²} = 2 + √(x² + y²)

Now, squaring both sides, we get

{(x - 2)² + (y - 2)²} = {2 + √(x² + y²)}²

⇒ {(x - 2)² + (y - 2)²} = 2² + {2 × 2 × √(x² + y²)}
+ {√(x² + y²)}²

⇒ x² - 4x + 4 + y² - 4y + 4 = 4 + 4√(x² + y²) + x² + y²

⇒ - 4x - 4y + 4 = 4√(x² + y²)

⇒ - x - y + 1 = √(x² + y²)

Again, squaring both sides, we get

(- x - y + 1)² = x² + y²

⇒ x² + y² + 1² + 2xy - 2x - 2y = x² + y²

⇒ 1 + 2xy - 2x - 2y = 0

⇒ 2xy +1 = 2(x + y),

which is the required locus of the mentioned points in the given question.

Thank you for your question.