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Thread: Parametrics question

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    Junior Member HeroWise's Avatar
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    Parametrics question

    P is any variable point on the parabola x^2=-4y. THe tangent from P cuts the parabola x^2=4y at Q and R. Show that 3x^2=4y is the equation of the licus of the mid point of the chord RQ.






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    617 pages fan96's Avatar
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    Re: Parametrics question

    use [ tex ] and [ /tex ] (remove the spaces inside the square brackets).

    For example, [ tex ] x [ /tex ] gives

    The equation of the tangent at any point on the parabola is given by:



    Solving simultaneously the equations of the tangent and the parabola , we get



    Treating this as a quadratic in , we can solve it using the quadratic formula to obtain:





    Taking the midpoint and simplifying gives us the parametric equation



    and it's easy to show that the equivalent Cartesian equation is .

    (A nice trick you can use to find the midpoint in questions like these is to halve the sum of roots - this is most useful when you don't need the coordinates.)
    Last edited by fan96; 30 Jun 2018 at 8:56 PM.
    HSC 2018 - [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design and Development]

    1(3√3) t2 dt cos(3π/9) = log(3√e) | Integral t2 dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.

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