Results 1 to 6 of 6

Math Help - locus of tangents

  1. #1
    Member
    Joined
    Nov 2010
    Posts
    112

    locus of tangents

    RHS show that the tangents at the extremities of any focal chord of a parabola y^2=4ax intersect at right angles at directrix.
    My work:
    Consider ends of focal chord as P( at_1^2,2at_1)and Q( at_2^2,2at_2) .
    now point of intersection of tangents are( at_1t_2 ,2a(t_1+t_2).now we know t_1t_2=-1 hence point of intersection are (-a,a(t_1+t_2)).clearly this lies on directrix.
    but my second method creates a problem .we know that x-co-ordinate of point of intersection of tangents at P and Q on the parabola is the GEOMETRIC MEAN of the x coordinate of P and Q and y coordinate is the ARITHMETIC MEAN of y coordinates(by definition of book) .therefore for point of intersection I take the GM of x coordinate as \sqrt( a^2t_1^2t_2^2) hence answer must be a (x coordinate)as a>0.
    Where is the error.is the definition of book is true for GM and AM for some special case.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116

    Re: locus of tangents

    Quote Originally Posted by ayushdadhwal View Post
    RHS show that the tangents at the extremities of any focal chord of a parabola y^2=4ax intersect at right angles at directrix.
    My work:
    Consider ends of focal chord as P( at_1^2,2at_1)and Q( at_2^2,2at_2) .
    now point of intersection of tangents are( at_1t_2 ,2a(t_1+t_2).now we know t_1t_2=-1 hence point of intersection are (-a,a(t_1+t_2)).clearly this lies on directrix.
    but my second method creates a problem .we know that x-co-ordinate of point of intersection of tangents at P and Q on the parabola is the GEOMETRIC MEAN of the x coordinate of P and Q and y coordinate is the ARITHMETIC MEAN of y coordinates(by definition of book) .therefore for point of intersection I take the GM of x coordinate as \sqrt( a^2t_1^2t_2^2) hence answer must be a (x coordinate)as a>0.
    Where is the error.is the definition of book is true for GM and AM for some special case.
    There isn't any error. You stopped only one step before the finish:

    Since t_1t_2=-1 the GM becomes \sqrt{ a^2t_1^2t_2^2} = \sqrt{a^2 \cdot (-1)^2}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23

    Re: locus of tangents

    There is a geometry argument without any algebra for this problem.
    Denote the two extremities of the focal chord as A and B, let M be the midpoint of the segment AB. Draw a segment MH which is parallel to the x-axis, crossing the directrix at H.
    Draw AD and BE parallel to the x-axis, crossing the directrix at D, E respectively. Connect AH and BH. Let F be the focal point.
    Then we have AB=AF+FB=AD+BE=2MH. So we have AM=MB=MH. So AHB is a right triangle. We need only to show that AH and BH are tangent lines. This is quite easy using the optical properties of the parabola.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Nov 2010
    Posts
    112

    Re: locus of tangents

    \sqrt{ a^2t_1^2t_2^2} = \sqrt{a^2 \cdot (-1)^2} sir here (-1)^2=1 so if one use this then answer is a .
    also sir GM is applicable only when numbers are positive .so how can we apply GM here when we are not sure about the coordinate that they are positive or negative .
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116

    Re: locus of tangents

    Quote Originally Posted by ayushdadhwal View Post
    \sqrt{ a^2t_1^2t_2^2} = \sqrt{a^2 \cdot (-1)^2} sir here (-1)^2=1 so if one use this then answer is a .
    also sir GM is applicable only when numbers are positive .so how can we apply GM here when we are not sure about the coordinate that they are positive or negative .
    The equation of the parabola is given as:

    y^2 = 4 \cdot a \cdot x

    If a > 0 then all x-values must be positive or zero because y^2 \ge 0.

    If a < 0 then all x-values must be negative or zero because y^2 \ge 0.

    In both cases the x-values have the same sign which is sufficient for the use of the GM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    lww
    lww is offline
    Newbie
    Joined
    Mar 2012
    From
    P. R. China
    Posts
    1

    Re: locus of tangents

    For two negtive number $x_1,x_2$, we can get $\sqrt{x_1x_2}$ of course.

    If some one believes that $\sqrt{x_1x_2}$ is the geometric mean of two negtive number $x_1,x_$, then we will get the conclusion that in some cases the "mean" is greater than every memeber. Then how can we call it the "mean"?

    If some one believes that $-\sqrt{x_1x_2}$ is the geometric mean of two negtive number $x_1,x_$, then how can we explain the inequality AM >= GM?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: April 19th 2010, 12:35 PM
  2. Equation of locus of two tangents
    Posted in the Geometry Forum
    Replies: 3
    Last Post: October 14th 2009, 06:30 PM
  3. Locus
    Posted in the Geometry Forum
    Replies: 1
    Last Post: June 5th 2009, 01:55 PM
  4. Locus
    Posted in the Geometry Forum
    Replies: 4
    Last Post: September 30th 2008, 02:37 PM
  5. Locus Help
    Posted in the Geometry Forum
    Replies: 34
    Last Post: July 28th 2007, 05:47 PM

Search Tags


/mathhelpforum @mathhelpforum