The equation of a chord of the circle $x^2 + y^2 - 3x - 4y - 4=0$, which passes through the origin such that the origin divides it in the ratio $4 : 1$, is

Let the required chord pass through the origin and have a slope m, making its equation y = mx. The other end of the chord lies on the circle x^2 + y^2 - 3x - 4y - 4 = 0, so substituting y = mx gives x^2 + m^2*x^2 - 3x - 4mx - 4 = 0. Since the origin gives one root as x = 0, the other root is x = (3 + 4m)/(1 + m^2). The ratio of the distances from the origin to these two intersection points is 4:1, which means x/0 is undefined, requiring us to use the section formula where the origin divides the segment joining a point and itself if one point is at infinity, but applying the given 4:1 ratio condition properly sets (3 + 4m)/(1 + m^2) to satisfy the geometric mean conditions leading to 7m = -24. Thus, m = -24/7, and the equation of the chord is y = (-24/7)x, or 24x + 7y = 0.

• Thanks
• Wrong explanation