The line x - y = 0 cuts the circle x² + y² + 2x = 0 at two points A and B. Find the co-ordinates of the points A and B. Also, find the equation of a circle whose diameter is AB.

Solution:
Given,
(i) x - y = 0
(ii) x² + y² + 2x = 0
Lines (i) and (ii) intersect each other at two points.

Solving
(i) x - y = 0
or, x = y

(ii) x² + y² + 2x = 0
or, x² + x² + 2x = 0
or, 2x² + 2x = 0
or, 2x(x + 1) = 0

Either,
2x = 0, x = 0
Or,
x +1 = 0, x = -1

Let the co-ordinates of point A be (x1, y1) and that of point B be (x2,y2).

When x = 0, y = 0. So, co-ordinates of point A is (x1,y1) = (0,0).

When x = -1, y = -1. So, co-ordinates of point B is (x2,y2) = (-1,-1).

Let AB be the diameter.
Now, equation of a circle in diameter form is given by,

(x - x1)(x - x2) + (y - y1)(y - y2) = 0
or, (x - 0){x - (-1)} + (y - 0){y - (-1)} = 0
or, x(x + 1) + y(y +1) = 0
or, x² + x + y² + y = 0
or, x² + y² + x + y = 0 is the required equation.

Hence, the required equation of the given circle is (x² +y² +x + y = 0).

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