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    The point on the circle x^2 + y^2 + 6x + 8y = 75 which is closest to the origin, is how far from the origin.

    My solution:

    (x + 3)^2 + (y + 4)^2 = 100

    Line from origin to circle can be constructed using the coordinates of the center of the circle and the origin.

    So y = 3/4x is the line.

    (x + 3)^2 + (3/4x + 4)^2 = 100

    x = 3.6621
    when x = 3.6624, then y = 4.102

    Then work out the distance using Pythagoras.


    However, this is wrong. Could you please assist?
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    its y=4/3x not y=3/4x g
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    try that you should get (3,4) as the point youre looking and distance 5
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    (Original post by muttonrolls)
    try that you should get (3,4) as the point youre looking and distance 5
    Thanks for your response, but I'm afraid that y = 4/3x is what I got for the line equation the first time, that's a typo in the above post.

    What I do next is I insert 4/3x into the circle equation, like so:

    (x + 3)^2 + (4/3x + 4)^2 = 100

    This leads me to the Quadratic 25/9 (x^2) + 26/3 (x) - 75...

    Which leads me to find the x as 3.6621, and y as 4.102.

    :P
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    Even after extensive experimentation, I can't seem to get the answer for this one.

    I don't want to include all the ways I've tried to approach the problem, as that would take up too much space.

    Could someone guide me in the right direction?
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    Hey! Just graph this and you are gonna discover that the circle has radius 10, centre(3,4) And the distance between center and radius is 5.
    Cheers
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    (Original post by Lordtangent)
    Hey! Just graph this and you are gonna discover that the circle has radius 10, centre(3,4) And the distance between center and radius is 5.
    Cheers
    Edit: I noticed I havent quite finished the logic but you get that , the minimum distance is 5 as its in a straight line.
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    (Original post by Lordtangent)
    Edit: I noticed I havent quite finished the logic but you get that , the minimum distance is 5 as its in a straight line.
    I did do a graph drawing, but from my drawing it looks to me that the closest point the origin isn't the straight line that leads upwards to 5, but instead the continuation of the line that passes through the origin and the center of the circle.

    To do an accurate diagram, I would need to use a compass. :P Is using a compass something you would recommend?
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    (Original post by UCASLord)
    I did do a graph drawing, but from my drawing it looks to me that the closest point the origin isn't the straight line that leads upwards to 5, but instead the continuation of the line that passes through the origin and the center of the circle.

    To do an accurate diagram, I would need to use a compass. :P Is using a compass something you would recommend?
    If a circle has radius R, center C, and the distance of C from the origin is r, then the closest point between circle and origin is a distance |R-r| from the origin. You should not be worrying about lines at all.
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    (Original post by UCASLord)
    The point on the circle x^2 + y^2 + 6x + 8y = 75 which is closest to the origin, is how far from the origin.

    My solution:

    (x + 3)^2 + (y + 4)^2 = 100

    Line from origin to circle can be constructed using the coordinates of the center of the circle and the origin.

    So y = 3/4x is the line.

    (x + 3)^2 + (3/4x + 4)^2 = 100

    x = 3.6621
    I'm assuming you used a calculator for this. It should be stating the obvious, but if you need to use a calculator in the MAT, you've gone wrong somewhere

    when x = 3.6624, then y = 4.102
    Note that for your values of x and y, x^2 = 6.6624^2 > 36, (y+4)^2 = 8.012^2 > 64, so (x+3)^2+(y+4)^2 is definitely *not* 100.

    It is usually easier to check solutions that to find them in the first place, and so checking is usually a good thing to do...
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    (Original post by UCASLord)
    Thanks for your response, but I'm afraid that y = 4/3x is what I got for the line equation the first time, that's a typo in the above post.

    What I do next is I insert 4/3x into the circle equation, like so:

    (x + 3)^2 + (4/3x + 4)^2 = 100

    This leads me to the Quadratic 25/9 (x^2) + 26/3 (x) - 75...
    (x+3)^2 = x^2 + 6x + 9
    (4x/3 + 4)^2 = 16x^2 / 9 + 32x/3 + 16 (***)

    so we end up with 25/9 x^2 + 50/3 x - 75 = 0.

    I'm guessing you somehow lost a factor of 4 in the 'x' term when squaring to find (***).
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    (Original post by DFranklin)
    If a circle has radius R, center C, and the distance of C from the origin is r, then the closest point between circle and origin is a distance |R-r| from the origin. You should not be worrying about lines at all.
    Hmm, that seems to give the distance of 3 (because 10 - 3 - 4 = 3), but the actual distance is 5.
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    (Original post by UCASLord)
    Hmm, that seems to give the distance of 3 (because 10 - 3 - 4 = 3), but the actual distance is 5.
    The circle has center (-3, -4) which is distance 5 from the origin. It has radius 10. The distance given by |R-r| is therefore |10 - 5| = 5.

    Edit: did you think the distance of (-3, -4) from the origin was 7?!?
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    (Original post by DFranklin)
    The circle has center (-3, -4) which is distance 5 from the origin. It has radius 10. The distance given by |R-r| is therefore |10 - 5| = 5.

    Edit: did you think the distance of (-3, -4) from the origin was 7?!?
    Yeah, I forgot to use Pythagoras :P I see it now, distance to the origin is 5 now, and then 5 to the point.

    Thanks :P
 
 
 
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