Show that the pair of tangents drawn from (g,f) to the circles x^2+y^2+2gx+2fy+c=0 are at right angles if g^2+f^2+c=0?

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  • 4 years ago
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    The eqn of the circle x^2+y^2+2gx+2fy+c=0 can be re-written as

    (x+g)^2+(y+f)^2=g^2+f^2-c

    therefore the centre of the circle is C(-g,-f) and its radius is r where

    r^2=g^2+f^2-c.

    Let the two tangent points be S and T. Assuming that the tangents at S and T are at right angles, then the points P(+g,+f), S, T and the centre C of the circle form a square of side r=CS=CT, because the angles SPT, PTC and PSC are all right angles, so angle SCT must also be a right angle.

    So with this assumption P must lie on a circle of radius PC=r√2 from C.

    Therefore

    PC^2=(g--g)^2+(f--f)^2=(r√2)^2=2r^2

    4g^2+4f^2=2(g^2+f^2-c)

    g^2+f^2+c=0.

    ie Assuming that PS and PT are at right angles, then g^2+f^2+c=0 must be true.

    The converse must also be true : if g^2+f^2+c=0 then the tangents PS and PT must be at right angles.

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