Question

If S = 0 is a circle and (x₁, y₁) is a point in the plane of the circle (not the centre), then the equation S₁ = 0 represents a straight line which is perpendicular to the line joining the centre and the point (x₁, y₁).

Answer 1

Let S ≡ x² + y² + 2gx + 2gy + c = 0

Therefore

is a first-degree equation in x and y and hence it represents a straight line. Also, since the slope of the line S₁ = 0 is −(g + x₁)/(f + y₁), it follows that it is perpendicular to the line joining (−g, −f ) and (x₁, y₁).