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Math Help - Question about circle

  1. #1
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    Can you help me?(Question about circle)

    The circle S has equation :


    X2+Y2- 2X-4Y=8
    Show thatA(-1,-1)and B (3,5) are ends of diameter


    Show that C(4,4)lies on S and determine angle ACB


    Find the distance from D(5,0) to the nearest point E on S



    Find the the x-coordinates of the points of intersection of S with the line joining D and F(0,2).
    Last edited by Diligent_Learner; December 2nd 2008 at 02:43 PM.
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  2. #2
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    Quote Originally Posted by Diligent_Learner View Post
    The circle S has equation :


    x^2+y^2- 2x-4y=8

    Show thatA(-1,-1)and B (3,5) are ends of diameter


    Complete the square in the equation for both x and y to locate the center. Then, find the midpoint of AB to see if they are one and the same. The center of the circle would be the midpoint of the diameter.

    (x^2-2x+1)+(y^2-4y+4)=8+5

    (x-1)^2+(y-2)^2=13

    Center is at (1, 2)

    Midpoint of AB = \left(\frac{3-1}{2},  \ \  \frac{5-1}{2}\right)={\color{red}(1, 2)}

    Quote Originally Posted by Diligent_Learner View Post
    Show that C(4,4)lies on S and determine angle ACB


    Substitute (4, 4) into your original equation to see if it makes a true statement

    4^2+4^2- 2(4)-4(4)=8??

    Angle ACB is an inscribed angle in a semicircle, therefore 90 degrees.



    Quote Originally Posted by Diligent_Learner View Post
    Find the distance from D(5,0) to the nearest point E on S


    See Earboth's post.

    Quote Originally Posted by Diligent_Learner View Post
    Find the the x-coordinates of the points of intersection of S with the line joining D and F(0,2).
    Define a linear equation for the line passing through D(5, 0) and F(0, 2). Then solve the system which includes this linear equation and the equaton of the circle. You should get two distinct points of intersection. Try it.
    Last edited by masters; December 5th 2008 at 09:32 AM.
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  3. #3
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    Thank you very much for your great help.I understand the question

    The equation of the line that passes through D and F is:

    y-y1=m(x-x1)


    Y= -0.4x+2

    ====================

    X2+y2-2x-4y=8

    X2+(-0.4x+2)2-2x-4(-0.4x+2)

    X2-3.6X-12=0


    X=5.7 or X=-2.1


    Last edited by Diligent_Learner; December 5th 2008 at 06:06 AM.
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  4. #4
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    Quote Originally Posted by Diligent_Learner View Post
    [FONT=Times New Roman]...

    Find the distance from D(5,0) to the nearest point E on S
    ...
    Let C denote the center of the circle. Then the Point E lies on the circle and on the line CD:

    CD: y =-\dfrac12 x + \dfrac52

    Calculating the intersection points of the circle and this line:

    (x-1)^2+\left(-\dfrac12 x + \dfrac52 - 2\right)^2 = 13

    yields: x = 1\pm \frac15 \sqrt{65} The x-coordinate of E must be greater than 1: Plug in this value into the equation of the line to get the y-coordinate. I've got E\left(1+\frac15\sqrt{65}\ ,\  2-\frac15\sqrt{65}\right)

    The distance ED = CD - r~\implies~ ED=\sqrt{20} - \sqrt{13} \approx 0.86658...
    Attached Thumbnails Attached Thumbnails Question about circle-nearestpointe.png  
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  5. #5
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    Thank you very much for clarifying the idea ..

    and Can yiu tell me about if my answer is true in the previous post?

    and in which program you draw the circle?
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  6. #6
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    added a sketch

    Quote Originally Posted by Diligent_Learner View Post
    Thank you very much for clarifying the idea ..

    and Can yiu tell me about if my answer is true in the previous post?

    and in which program you draw the circle?
    You've got the equation of the line correctly.
    But I've got results, different from yours, calculating the points of intersection:

    l:y=-\dfrac25 x+2

    Plug in the term of y into the equation of the circle:

    (x-1)^2+\left(-\dfrac25 x+2 - 2\right)^2=13

    x^2-2x+1+\dfrac4{25}x^2=13

    \dfrac{29}{25} x^2 - 2x-12=0

    Use the quadratic formula to solve this equation:

    x=\dfrac{25}{29} \pm \dfrac5{29} \sqrt{373}

    Take these x-values and plug them into the equation of the line to get the y-coordinates of the points of intersection.

    EDIT: Use the attached sketch to control your results.
    Attached Thumbnails Attached Thumbnails Question about circle-schnittkreis_gerade.png  
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