If y + 3x = 0 is the equation of a chord of the circle, x² + y² - 6x = 0, then the equation of the circle with this chord as diameter is :

Given y + 3x = 0 is the equation of a chord of the circle, x² + y² - 6x = 0
Solving given equation we get y = -3x
(x)² + (-3x)² - 6x = 0
=> 10x² - 6x = 0
=> x(10x - 6) = 0
=> x = 0 and x = 3/5
The points of intersection are (0, 0) and (3/5, -9/5)
The equation of the circle with the chord as diameter is given by:
(x - x1)(x - x2) + (y - y1)(y - y2) = 0
where (x1, y1) = (0, 0) and (x2, y2) = (3/5, -9/5)
(x - 0)(x - 3/5) + (y - 0)(y + 9/5) = 0
=> x² - (3/5)x + y² + (9/5)y = 0
Multiplying by 5, we get
5x² - 3x + 5y² + 9y = 0
=> 5x² + 5y² - 3x + 9y = 0