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Math Help - complex no

  1. #1
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    complex no

    if z1=10+6i and z2=4+6i if z is any complex no such that the argument of (z-z1)/(z-z2)
    is pi/4 then prove that mod(z-7-9i)=3sqrt(2)

    can yu solve it graphically
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  2. #2
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    Quote Originally Posted by prasum View Post
    if z1=10+6i and z2=4+6i if z is any complex no such that the argument of (z-z1)/(z-z2)
    is pi/4 then prove that mod(z-7-9i)=3sqrt(2)

    can you solve it graphically
    Graphically, the argument of the quotient (z-z_1)/(z-z_2) is the angle between the lines z\to z_1 and z\to z_2. By the theorem about angles in the same segment (or rather its converse), that says that z is on a circle through z_1 and z_2. Another theorem (the one about the angle at the centre C being twice the angle at the circumference) says that the lines z_1\to C and z_2\to C must be at right angles. It's easy to deduce from that (see the diagram) that C must be at the point 7+9i and that the radius must be 3\sqrt2. So z lies on the circle |z-(7+9i)| = 3\sqrt2.
    Attached Thumbnails Attached Thumbnails complex no-circle.png  
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