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    Hello,

    Could anybody please help me with how to tackle fp3 locus questions.

    There is a question in the FP3 edexcel book, which mentions:

    "the tangent to the ellipse with equation x^2/a^2 + y^2/b^2 = 1 at the point P(acost, bsint) crosses the x-axis at A and the y-axis at B.

    Find an equation for the locus of the mid-point of AB as P moves round the ellipse."

    I have completed the question with the help of the book, however I cannot comprehend the second part of the question: "for the locus of the mid-point of AB as P moves round the ellipse": what does this mean?

    Thank you
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    It means the path that the mid point of AB takes as the point P varies. The equation that the question asks you for describes this path.
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    it means you work out the coordinates of the mid point of AB and then combine them to remove the parameter used (also isnt that an example in the book :P)
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    This question requires good understanding of general points.

    First of all you need to find the gradient of the tangent at the point (acost,bsint). This can be done using the rule:-
     \frac{dy}{dx} = \frac {\frac {dy}{dt}}{\frac {dx}{dt}}

     x = acos(t), y = bsin(t) therefore:-

     \frac {dy}{dx} = \frac {-bcos(t)}{asin(t)} .

    Now we use the formula for the equation of a line to find out the equation of the tangent

     y - bsin(t) = \frac {-bcos(t)}{asin(t)}(x - acos(t)) which simplifies to

     aysin(t) + bxcos(t) = 2absin(t)cos(t)

    Point A is where the tangent crosses the x-axis i.e. when y = 0

     x = 2asin(t), A = (2asin(t) , 0)

    Point B is where the tangent crosses the y-axis i.e. when x = 0

     y = 2bcos(t) B = (0 , 2bcos(t))


    aaaaaand just realised I didn't need to do all that lol. The locus is basically all the positions that it possibly take. It seems daunting to try and work it out but if you follow the geometric rules for when you have actual values of x and y and you just use the general point instead, instead of just getting one point you get a set of points, or loci, which just so happens to be an equation of another shape!
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    (Original post by Phil_Waite)
    It means the path that the mid point of AB takes as the point P varies. The equation that the question asks you for describes this path.

    So does the chord AB move as P moves?
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    (Original post by sulexk)
    So does the chord AB move as P moves?
    Yes, as it is a tangent to the ellipse at that point, so as the point moves, the tangent moves with it.
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    (Original post by Phil_Waite)
    Yes, as it is a tangent to the ellipse at that point, so as the point moves, the tangent moves with it.
    There are often questions on finding the locus equation. There have been many times where finding the starting point of answering these questions, seems very difficult. Any ideas?


    Thank you so much!
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    (Original post by sulexk)
    There are often questions on finding the locus equation. There have been many times where finding the starting point of answering these questions, seems very difficult. Any ideas?


    Thank you so much!
    Well always make sure you draw a diagram so that you can see graphically what information the question has given you, and what you need to work out. Then the way to start working it out would be to do something that is related to the type of locus that it is, so if it is a tangent, then you first need to find what its gradient would be, so you'd differentiate the equation of the curve it is tangential to. If it is a fixed distance from a point, then you'd square some x and y coordinates, etc.
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    (Original post by Phil_Waite)
    Well always make sure you draw a diagram so that you can see graphically what information the question has given you, and what you need to work out. Then the way to start working it out would be to do something that is related to the type of locus that it is, so if it is a tangent, then you first need to find what its gradient would be, so you'd differentiate the equation of the curve it is tangential to. If it is a fixed distance from a point, then you'd square some x and y coordinates, etc.
    Thank you.

 
 
 
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