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    Find the cartesian equation of the locus of the set of points P if
    P is equidistant from the lines 3x+4y+5 = 0 and 12x-5y+13 = 0

    So i tried to get 2 set of points from these lines by picking two different x and then finding their corresponding y.
    I named the point P (x,y) and the other two points A and B respectively.
    If P is equidistant from A and B, then PA=BA
    Using the distance formula for (PA)^2= (PB)^2, i got an equation in terms of y and x.
    However the textbook gave 2 eqns and none of them match the one i got.
    Any problems in the approach im using?
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    (Original post by Gopee)
    P is equidistant from the lines 3x+4y+5 = 0 and 12x-5y+13 = 0

    So i tried to get 2 set of points from these lines by picking two different x and then finding their corresponding y.
    I named the point P (x,y) and the other two points A and B respectively.
    If P is equidistant from A and B, then PA=BA
    Using the distance formula for (PA)^2= (PB)^2, i got an equation in terms of y and x.
    However the textbook gave 2 eqns and none of them match the one i got.
    Any problems in the approach im using?
    You picked a single point on each of the two given lines. What makes you think that the distance between them and your point P must always be equal?

    This is the scenario as I see it:

    Spoiler:
    Show




    where the blue and red lines represent the ones in the question, and the dashed orange/green lines are the ones that you're trying to find. They are in fact the lines of symmetry for the two lines in question.

    In your working, you might've picked one point very far along the red line and another very close to the their point of intersection. Their distances to the orange/green lines are definitely not the same, you got nothing that guarantees it.

    Have another think.
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    ok the way i see it, for the points to be equidistant from p, they need to be a reflection from one of the other where the Equation of line P being the line of reflection, So one point of x will have 2 sets of reflection , when when p is nearly horizontal and the other when p is nearly vertical. How do i find the reflected points of x?
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    Is that an a level question? I never came across one of these so far....
    As for the question, i would have use silmultaneous equation to find the point of intersection , choose another set of random points that lies on any of the two given lines , name a third set of points P which forms a right angle triangle together with the 2 previous points. Use the distance formula to create two equations with P as unknown coords, then use silmultaneous eqn again to find P.
    Then use the now known points of P and the previous point of intersection to calculate gradient and the equation of line. Also since the gradient is now known, you can calculate the normal to the line on which p lies.

    Is that a valid method?
 
 
 
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