Question 45

The points in the xy-plane, which satisfy the equation
$$\sqrt{(x - 1)^2 + (y + 2)^2} = \sqrt{(x + 3)^2 + (y - 2)^2}$$

Solution

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We are given the equation, $$\sqrt{(x - 1)^2 + (y + 2)^2} = \sqrt{(x + 3)^2 + (y - 2)^2}$$

Squaring on both sides, we get,

$$(x-1)^2+(y+2)^2\ =\ (x+3)^2+(y-2)^2$$

$$x^{2\ }+\ 1\ -\ 2x\ +\ y^2\ +\ 4\ +\ 4y\ =\ x^{2\ }+\ 9\ +\ 6x\ +\ y^2\ +\ 4\ -\ 4y\ $$

$$8y\ =\ 8x\ +\ 8$$

$$y\ =\ x\ \ +\ 1$$

The above statement represents a straight line.

Hence, the correct answer is option A.

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