The equation of the circle described on the chord $3x+y+5=0$ of the circle $x^2+y^2=16$ as diameter is :

Let the required circle be S = x^2 + y^2 - 16 + k(3x + y + 5) = 0, which simplifies to x^2 + y^2 + 3kx + ky + 5k - 16 = 0 with center at (-3k/2, -k/2). The chord 3x + y + 5 = 0 must be a diameter, so the center lies on the chord. Substituting the center into the chord equation gives 3(-3k/2) + (-k/2) + 5 = 0, which simplifies to -5k = -5, so k = 1. Substituting k = 1 into the circle equation yields x^2 + y^2 + 3x + y - 11 = 0.

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