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  1. #1
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    Circle question

    A moving point P is such that the length of the tangent from P to the circle x^2+y^2=16 is equal to the distance of P from point (8,8). Show that the locus of P is the straight line x+y=9.

    Can anyone help me? Thanks...
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  2. #2
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    Quote Originally Posted by MichaelLight View Post
    A moving point P is such that the length of the tangent from P to the circle x^2+y^2=16 is equal to the distance of P from point (8,8). Show that the locus of P is the straight line x+y=9.

    Can anyone help me? Thanks...
    1. Draw a sketch!

    2. Let P(x, y) denote a point whose distance to Q(8, 8) equals the tangent segment to the given circle. The you know that the distance d can be calculated by:

    d^2 = (x-8)^2+(y-8)^2

    3. The distance d, the radius r of the circle and the distance of P to the center of the circle form a right triangle:

    d^2+r^2=x^2+y^2

    4. Plug in r = 4 and dē from the 1st equation into the 2nd equation:

    (x-8)^2+(y-8)^2+16=x^2+y^2

    Solve for y.
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  3. #3
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by MichaelLight View Post
    A moving point P is such that the length of the tangent from P to the circle x^2+y^2=16 is equal to the distance of P from point (8,8). Show that the locus of P is the straight line x+y=9.

    Can anyone help me? Thanks...
    draw a circle C of radius 0 at the point (8,8). Then the distance of P from (8,8) is the length of the tangent drawn from P to C.
    So the locus of P is the radical axis of the circles {(x-8)}^2+{(y-8)}^2=0 and x^2+y^2=16.
    we know that radical axis of any two cirlces is a straight line.
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