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Math Help - Another question on Circle

  1. #1
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    Another question on Circle

    I have no idea to deal with this question. Can anyone give me some hints?

    A circle passes through the point A(2, -1) and is tangent to the line y= x - 1. Its centre is on the line 2x + y = 0. Find the equation of the circle.
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  2. #2
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    The red line is the tangent line.
    The green line is where the center lies on.
    A circle dot represents a point which lies on the circle.
    A rhombus dot represents the center of circle.


    This is what I did.
    1)Drew the red line and green line and circle dot (given to us).
    2)Pick a point (x,y) on the green line and find locus that is equidistanct from the intersection point (1/3,-2/3) and the dot (2,-1).
    3)The locus was a line (not show), it intersected the green line at (1,-2)-the rhombus dot, and that was the center of circle.
    4)Now we have a center (1,-2) and a point on it (2,-1) we can find the distance, sqrt(2), and find the equation of circle.
    (x-1)^2+(y+2)^2=2
    Attached Thumbnails Attached Thumbnails Another question on Circle-picture2.gif  
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  3. #3
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    Hello, ling_c_0202!

    I can give you some hints . . .


    A circle passes through the point A(2, -1) and is tangent to the line L: y = x - 1.
    Its centre is on the line M: 2x + y = 0.
    Find the equation of the circle.
    It certainly helps to make a sketch.
    Of course, we want the center and radius of the circle.


    The center is on line M: .y = -2x
    So the center C has coordinates: (p, -2p)

    Then the distance from C to A is the radius.
    . . sqrt{9p - 2) + (-2p + 1)} .= .r [1]

    You need to find the perpendicular distance from C to line L.
    . . That too is the radius r.
    When you find that, you'll have another equation in p and r. [2]
    . . Solve the system of equations.

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    I have found two solutions.
    . . . . . . . . . . . . _
    . . C(1, -2), r = √2
    . . . . . . . . . . . . . . ._
    . . C(9, -18), r = 13√2

    And they both check out. .Anyone care to double-check them?

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  4. #4
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    Quote Originally Posted by Soroban View Post
    [size=3]
    . . sqrt{9p - 2) + (-2p + 1)} .= .r [1]
    Thank you very much! I have got the answers!
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