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Math Help - equations intersections

  1. #1
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    equations intersections

    hi, my question is: find the equations of the tangents to the curve with equation y = (x + 1)(5 - x) at the points P(-1, 0) and Q(5, 0). prove that these two tangents intersect at the point (2, 18)

    the first thing i did was expand the brackets and differentiate with x = -1 (for P). which resulted in:

    -x^2 + 4x + 5 so then \frac{dy}{dx} = - 2x + 4 and usig x = -1 the gradient comes to 6

    so the equation is y - 0 = 6(x + 1) \implies y = 6x + 6

    so for Q i did the same thing. used x = 5 for -2x + 4 which gave a gradient of - 6

    so the equation is y - 0 = -6(x - 5) \implies y = -6x + 30


    how would i prove these two tangents intersect at point (2, 18)?

    thanks, mark
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  2. #2
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    Quote Originally Posted by mark View Post
    hi, my question is: find the equations of the tangents to the curve with equation y = (x + 1)(5 - x) at the points P(-1, 0) and Q(5, 0). prove that these two tangents intersect at the point (2, 18)

    the first thing i did was expand the brackets and differentiate with x = -1 (for P). which resulted in:

    -x^2 + 4x + 5 so then \frac{dy}{dx} = - 2x + 4 and usig x = -1 the gradient comes to 6

    so the equation is y - 0 = 6(x + 1) \implies y = 6x + 6

    so for Q i did the same thing. used x = 5 for -2x + 4 which gave a gradient of - 6

    so the equation is y - 0 = -6(x - 5) \implies y = -6x + 30


    how would i prove these two tangents intersect at point (2, 18)?

    thanks, mark
    Solve simultaneously:

    y = 6x + 6 .... (1)

    y = -6x + 30 .... (2)

    And in case you're wondering, I Thanked your post because you showed all your working and you clearly stated where you got stuck (a rarity for this kind of question).
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  3. #3
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    ah thanks. i really should have thought of solving them simultaneously. thanks for the help
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  4. #4
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    actually hold on, i tried simultaneously solving them and it didn't turn out how i expected.

    so its 6x + 6 = -6x + 30 which leaves y = 12x - 24

    or i just added equation 1 and 2 together to cancel the x's and it left y = 36

    could someone show me what to do from here please? my maths has gotten really rusty from not doing it for ages

    thanks
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  5. #5
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    ah don't worry i got it now
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