Results 1 to 3 of 3
Like Tree2Thanks
  • 2 Post By Idea

Thread: Curve were all points lie just as far from P as from m.

  1. #1
    Newbie
    Joined
    Apr 2019
    From
    Norway
    Posts
    1

    Curve were all points lie just as far from P as from m.

    Hi, I am stuck on this task;
    Thanks in advance


    A curve (c) goes so that all points lie just as far from P as from m.
    P(1,3) and M(x)= 2x-4


    What type of curve is c?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    1,017
    Thanks
    519

    Re: Curve were all points lie just as far from P as from m.

    this is a parabola by definition

    Curve were all points lie just as far from P as from m.-parabola.png
    Last edited by Idea; Apr 30th 2019 at 09:22 AM.
    Thanks from topsquark and HallsofIvy
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    20,249
    Thanks
    3366

    Re: Curve were all points lie just as far from P as from m.

    Let $(x_0, y_0)$ be a point on this curve. Its distance from (1, 3) is $\sqrt{(x_0- 1)^2+ (y_0- 3)^2}$. Its distance from y= 2x- 4 is $\frac{2x_0- y_0- 4}{\sqrt{5}}$. Those two distances will be the same if and only if $\sqrt{(x_0- 1)^2+ (y_0- 3)^2}= \frac{2x_0- y_0- 4}{\sqrt{5}}$. Squaring both sides, $(x_0- 1)^2+ (y_0- 3)^2= \frac{2x_0- y_0- 4)^2}{5}$. Multiplying everything, $5x_0^2- 10x_0+ 5+ 5y_0^2- 30y_0+ 45= 4x_0^2+ y_0^2+ 16- 4x_0y_0- 16x_0+ 8y_0$.

    $x_0^2+ 4x_0y_0+ 4y_0^2+ 6x_0- 38y_0+ 29= 0$.

    The second degree terms are of the form "$Ax^2+ Bxy+ Cy^2$ with A= 1, B= 4, and C= 4. It's "discriminant" is $B^2- 4AC= 16- 16= 0$ which tells us this conic section is a parabola.

    As Idea said, one of the definitions of a parabola is "a curve that is always equi-distant from a point (the "focus") and a line (the "directrix").
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Aug 24th 2011, 11:35 AM
  2. closes points to points on the curve
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Jun 13th 2011, 10:24 PM
  3. points on a curve
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Feb 15th 2010, 11:04 AM
  4. finding points on the curve
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Jan 19th 2010, 10:21 AM
  5. Replies: 10
    Last Post: Jan 16th 2009, 09:19 PM

/mathhelpforum @mathhelpforum