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Heinemann Pure5 Exercise 4C Questions 17 and 20 watch

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    Evening all. Just finished Ex 4c but for two questions... Any help greatly appreciated as always.

    17.
    The normal at the point P(2t,2/t) to the curve with equation xy=4 meets the lines with equations y=x and y=-x at the points Q and R repectively. Prove that PQ=PR.

    I managed to get it down to two very similiar, but not equal, lengths in t....

    20.
    The tangents at P(ct1,c/t1) and Q(ct2,c/t2) to the rectangular hyperbola with equation xy=c^2 meet on the rectangular hyperbola with equation xy=c^2/4. Prove that PQ is a tangent to the curve with equation xy=4c^2.

    I'm not sure how the seperate parts of the questions relate.

    Thanks!
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    (17)
    On the curve, y + x dy/dx = 0. So the tangent at P has gradient -(2/t)/(2t) = -1/t^2. The normal at P has gradient t^2 and equation y = t^2 (x - 2t) + 2/t.

    At Q,

    t^2 (x - 2t) + 2/t = x
    x (t^2 - 1) = 2t^3 - 2/t = (2/t)(t^4 - 1) = (2/t)(t^2 - 1)(t^2 + 1)
    x = (2/t)(t^2 + 1) = 2t + 2/t
    y = 2t + 2/t

    So

    |PQ|^2
    = (2t + 2/t - 2t)^2 + (2t + 2/t - 2/t)^2
    = 4/t^2 + 4t^2

    At R,

    t^2 (x - 2t) + 2/t = -x
    x (t^2 + 1) = 2t^3 - 2/t = (2/t)(t^4 - 1) = (2/t)(t^2 - 1)(t^2 + 1)
    x = (2/t)(t^2 - 1) = 2t - 2/t
    y = -2t + 2/t

    So

    |PR|^2
    = (2t - 2/t - 2t)^2 + (-2t + 2/t - 2/t)^2
    = 4/t^2 + 4t^2
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    (Original post by deity47)
    ...

    20.
    The tangents at P(ct1,c/t1) and Q(ct2,c/t2) to the rectangular hyperbola with equation xy=c^2 meet on the rectangular hyperbola with equation xy=c^2/4. Prove that PQ is a tangent to the curve with equation xy=4c^2.

    I'm not sure how the seperate parts of the questions relate.

    Thanks!
    You use the first part about the tangents meeting to define a relationship beytween t1 and t2.
    THEN you do the other bit. Get eqn of the line PQ and interesect it with the other hyperbola.
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    (Original post by Jonny W)
    2t^3 - 2/t = (2/t)(t^4 - 1)
    Cheers Jonny, that factorisation made it a lot simpler. Nice job.

    (Original post by Fermat)
    You use the first part about the tangents meeting to define a relationship beytween t1 and t2.
    THEN you do the other bit. Get eqn of the line PQ and interesect it with the other hyperbola.
    Thanks Fermat, I used the relationship to prove that the discriminant when PQ and the hyperbola intersect was zero, and then made sure that their gradients were equal.

    Thanks again! :tsr:
 
 
 
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