To find the number of points of intersection of the line and the circle, we need to substitute the equation of the line into the equation of the circle and solve for the variables.

Step 1: The equation of the line is x - y + 1 = 0. We can express y in terms of x as y = x + 1.

Step 2: Substitute y = x + 1 into the equation of the circle 3x^2 + 3y^2 + x + my - 3 = 0. This gives us 3x^2 + 3(x + 1)^2 + x + m(x + 1) - 3 = 0.

Step 3: Simplify the equation to get 6x^2 + (3 + m)x + 3m = 0.

Step 4: This is a quadratic equation in the form ax^2 + bx + c = 0. The discriminant of this equation is b^2 - 4ac. The number of points of intersection of the line and the circle depends on the value of the discriminant.

- If the discriminant is greater than 0, the line and the circle intersect at two distinct points.
- If the discriminant is equal to 0, the line and the circle intersect at exactly one point.
- If the discriminant is less than 0, the line and the circle do not intersect.

Step 5: Calculate the discriminant: (3 + m)^2 - 4*6*3m = m^2 - 24m + 9.

Step 6: The number of points of intersection of the line and the circle depends on the value of m. For different values of m, the discriminant will have different values, and hence the number of points of intersection will vary.

Question

To find the number of points of intersection of the line and the circle, we need to substitute the equation of the line into the equation of the circle and solve for the variables.

Step 1: The equation of the line is x - y + 1 = 0. We can express y in terms of x as y = x + 1.

Step 2: Substitute y = x + 1 into the equation of the circle 3x^2 + 3y^2 + x + my - 3 = 0. This gives us 3x^2 + 3(x + 1)^2 + x + m(x + 1) - 3 = 0.

Step 3: Simplify the equation to get 6x^2 + (3 + m)x + 3m = 0.

Step 4: This is a quadratic equation in the form ax^2 + bx + c = 0. The discriminant of this equation is b^2 - 4ac. The number of points of intersection of the line and the circle depends on the value of the discriminant.

- If the discriminant is greater than 0, the line and the circle intersect at two distinct points.
- If the discriminant is equal to 0, the line and the circle intersect at exactly one point.
- If the discriminant is less than 0, the line and the circle do not intersect.

Step 5: Calculate the discriminant: (3 + m)^2 - 4*6*3m = m^2 - 24m + 9.

Step 6: The number of points of intersection of the line and the circle depends on the value of m. For different values of m, the discriminant will have different values, and hence the number of points of intersection will vary.

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