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Math Help - coordinate geometry

  1. #1
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    coordinate geometry

    find the locus of centres of circle x^2+y^2-2ax-2by+2=0 where a and b are parameters if the tangents from any to each of the circles are orthogonal
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  2. #2
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    Hello, prasum!

    I'm not sure I understand the problem . . .


    Find the locus of centres of circle: x^2+y^2-2ax-2by+2\:=\:0
    . . where a and b are parameters,
    if the tangents from any point to each of the circles are orthogonal.

    The circle has the equation: . (x-a)^2 + (y-b)^2 \:=\:a^2+b^2-2

    It has center C(a,b) and radius r\,=\,\sqrt{a^2+b^2-2}

    Let P(h,k) be any point (exterior to the circle).

    Tangents are drawn from P to A and B on the circle.
    . . The tangents are orthogonal: . \angle P = 90^o


    Code:
        |
        |         * * *
        |     *           * A
        |   *               ♥
        |  *              o  *o
        |               o       o
        | *       C   o       *   o    P
        | *    (a,b)♥         *     ♥(h,k)
        | *           o       *   o
        |               o       o
        |  *              o  *o
        |   *               ♥
        |     *           * B
        |         * * *
    - - + - - - - - - - - - - - - - - - - - - - - -
        |

    Since \angle A = \angle B = \angle P = 90^o,\;PA = PB,\:CA = CB
    . . then quadrilateral APBC is a square.

    Its diagonal is: . CP \,=\,\sqrt{2}\,r \;=\;\sqrt{2(a^2+b^2-2)}


    Therefore, the locus of the centers of the circles
    . . is a circle with center (h,k) and radius \sqrt{2(a^2+b^2-2)}

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