the equation of a straight line l and a circle c are x-y+1=0and 3x^2+3y^2+x+my-3=0 respectively.find the number of points of intersaction of l and c

Step 1: Simplify the given equations

The equation of the line l can be rewritten as y = x + 1.

The equation of the circle c can be simplified by dividing through by 3 to get x^2 + y^2 + x/3 + my/3 - 1 = 0.

Step 2: Substitute y in the equation of the circle

Substitute y = x + 1 from the line equation into the circle equation to get x^2 + (x + 1)^2 + x/3 + m(x + 1)/3 - 1 = 0.

Step 3: Simplify the equation

Simplify the equation to get 2x^2 + (2/3 + 2m/3)x + 1 + m/3 - 1 = 0.

Step 4: Solve for x

The equation is a quadratic equation in the form ax^2 + bx + c = 0. The solutions for x are given by the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a).

Step 5: Determine the number of intersection points

The number of solutions for x corresponds to the number of intersection points between the line and the circle. If the discriminant b^2 - 4ac is positive, there are two solutions for x, meaning the line intersects the circle at two points. If the discriminant is zero, there is one solution for x, meaning the line is tangent to the circle and intersects it at one point. If the discriminant is negative, there are no real solutions for x, meaning the line does not intersect the circle.