Results 1 to 4 of 4

Math Help - analytic geometry in a plane - cirle

  1. #1
    Junior Member
    Joined
    Jan 2011
    Posts
    47

    analytic geometry in a plane - cirle

    find the equation of the circle that passes through the point ( 9,7) and is tangent to both the y-axis and line 3x-4y-24=0.


    What would be the best way to answer this question? can you please give me an illustration?
    I'm just really stuck on this one.
    Thanks in advance everyone.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by jam2011 View Post
    find the equation of the circle that passes through the point ( 9,7) and is tangent to both the y-axis and line 3x-4y-24=0.


    What would be the best way to answer this question? can you please give me an illustration?
    I'm just really stuck on this one.
    Thanks in advance everyone.
    1. Draw a sketch!

    2. The equation of the circle is

    (x-x_M)^2+(y-y_M)^2=r^2

    3. x_M=r Why?

    4. The point P(9, 7) is located on the circle-line. That means:

    (9-x_M)^2+(7-y_M)^2=r^2

    5. The distance of the midpoint M of the circle to the given line is r:

    \left| \dfrac{3x_M-4y_M-24}{\sqrt{3^2+4^2}} \right|=r

    6. Solve for x_M, y_M, r. There are two valid solutions.
    Attached Thumbnails Attached Thumbnails analytic geometry in a plane - cirle-tangachsenpkt.png  
    Last edited by earboth; March 7th 2011 at 10:29 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    rcs
    rcs is offline
    Senior Member rcs's Avatar
    Joined
    Jul 2010
    From
    iligan city. Philippines
    Posts
    455
    Thanks
    2

    interesting

    i love to solve this but what formula can i use? hhhmmm ..tnx
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by rcs View Post
    i love to solve this but what formula can i use? hhhmmm ..tnx
    1. You can't use a single formula to solve the question but you have to solve a system of equations:

    \left|\begin{array}{rcl}(9-x_M)^2+(7-y_M)^2&=&r^2 \\ x_M&=&r \\ \left|\dfrac{3x_M-4y_M-24}5 \right|&=&r \end{array}\right.

    2. The 3rd equation has to be transformed into 2 different equations:

    \left|\dfrac{3x_M-4y_M-24}5 \right|=r~\implies~\left\{\begin{array}{l}\dfrac{3  x_M-4y_M-24}5 =r \\ \dfrac{3x_M-4y_M-24}5  =-r\end{array}\right.

    3. To shorten the calculations: Use the equation with the negative sign. Then you'll get:

    \left|\begin{array}{rcl}(9-x_M)^2+(7-y_M)^2&=&r^2 \\ x_M&=&r \\ \left|\dfrac{3x_M-4y_M-24}5 \right|&=&r \end{array}\right.~\implies~(9-x_M)^2+(7-(2x_M-6))^2=x_M^2

    4. This is a simple quadratic equation which I leave for you. Have fun!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Analytic Geometry
    Posted in the Algebra Forum
    Replies: 3
    Last Post: February 18th 2010, 09:49 AM
  2. analytic, finite plane
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 11th 2009, 06:50 PM
  3. Analytic geometry help?
    Posted in the Geometry Forum
    Replies: 7
    Last Post: July 4th 2009, 11:42 AM
  4. Analytic Geometry Q2
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: January 30th 2009, 04:31 AM
  5. analytic geometry
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: January 6th 2009, 02:19 PM

Search Tags


/mathhelpforum @mathhelpforum