The equation of the circle described on the chord 3x+y+5=0 of the circle x2+y2=16 as diameter is:
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Official solution
Family of circles through the intersection of x2+y2−16=0 and the chord 3x+y+5=0:
\[x^2+y^2-16+\lambda(3x+y+5)=0.\]
Centre: (−23λ,−2λ). For the chord to be a diameter, the centre must lie on the chord line:
\[3\left(-\tfrac{3\lambda}{2}\right)+\left(-\tfrac{\lambda}{2}\right)+5=0\Rightarrow -5\lambda+5=0\Rightarrow\lambda=1.\]
Required circle: x2+y2+3x+y−11=0.