Results 1 to 4 of 4

Math Help - Hard Parametrics Problem

  1. #1
    Senior Member
    Joined
    Jul 2008
    Posts
    347

    Exclamation Hard Parametrics Problem

    P(2ap, ap^2) and Q(2aq,aq^2) are 2 points on the parabola x^2 = 4ay. Tangents to the parbaola at P and Q intersect at the point T.

    a) Show that the equation of the tangent at P is y=px-ap^2.
    b) Find the coordinates of T.
    c) P and Q move on the parabola so that the line PQ passes through the point (2a,-a). Show that p + q + 1 = pq.
    d) Hence, by finding the Cartesian equation of the locus T, show that T lies on a straight line.
    e) With the aid of a diagram, carefully explain why the locus of T is not all of the straight line.


    Could someone please show me how to do (e). I've found out that the locus of T is x-y+a = 0 and drawn myself a diagram with all the info provided and obtained but I have no clue how to use these to prove (e).
    Attached Thumbnails Attached Thumbnails Hard Parametrics Problem-parametrics.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello xwrathbringerx
    Quote Originally Posted by xwrathbringerx View Post
    P(2ap, ap^2) and Q(2aq,aq^2) are 2 points on the parabola x^2 = 4ay. Tangents to the parbaola at P and Q intersect at the point T.

    a) Show that the equation of the tangent at P is y=px-ap^2.
    b) Find the coordinates of T.
    c) P and Q move on the parabola so that the line PQ passes through the point (2a,-a). Show that p + q + 1 = pq.
    d) Hence, by finding the Cartesian equation of the locus T, show that T lies on a straight line.
    e) With the aid of a diagram, carefully explain why the locus of T is not all of the straight line.


    Could someone please show me how to do (e). I've found out that the locus of T is x-y+a = 0 and drawn myself a diagram with all the info provided and obtained but I have no clue how to use these to prove (e).
    The tangents cannot intersect at a point 'inside' the parabola, hence the locus is those segments of the straight line that lie 'outside' the parabola.

    Grandad
    Last edited by Grandad; January 23rd 2010 at 06:28 AM. Reason: Revised answer.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2009
    Posts
    54
    How about when P and Q are the same points Grandad?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello Sunyata
    Quote Originally Posted by Sunyata View Post
    How about when P and Q are the same points Grandad?
    So - what about it? What is the point you are making?

    It is meaningless to talk about the point of intersection of two tangents if the tangents are drawn at one and the same point. In the algebra, the point of intersection of the tangents at P and Q is found by solving the equation:
    (p-q)x=a(p^2-q^2)
    =a(p-q)(p+q)
    We can divide both sides by (p-q) to get the solution:
    x=a(p+q)
    only if p\ne q; otherwise we should be dividing by zero - and that's not allowed!

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Parametrics word problem
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: May 11th 2009, 02:03 PM
  2. Parametrics Help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 10th 2009, 12:17 PM
  3. Parametrics.....
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 25th 2009, 04:22 PM
  4. Parametrics
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: November 17th 2008, 01:40 PM
  5. Parametrics
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: April 8th 2008, 09:12 PM

Search Tags


/mathhelpforum @mathhelpforum