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Thread: [SOLVED] Anyone know ?

  1. #1
    jayson81
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    [SOLVED] Anyone know ?

    The curves y + ax^2 +b and y = 2x^2 +cx have a common tangent at the point (-1,0).
    Find a, b and c.
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  2. #2
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    Quote Originally Posted by jayson81
    The curves y + ax^2 +b and y = 2x^2 +cx have a common tangent at the point (-1,0).
    Find a, b and c.
    Firstly, (-1,0) is a common point thus.
    Substitute that into the second function,
    $\displaystyle 0=2(-1)^2+c(-1)$
    Thus,
    $\displaystyle 0=2-c$ solve, $\displaystyle c=2$
    Thus, the second function has equation,
    $\displaystyle y=2x^2+2x$.

    Since we say that these two function have a common tangent means the equation of the tangent through point (-1,0) is the same. Remember from analytic geomtery that two lines having same slope at same point are the same. Same thing here the tangent lines share a point thus they need to have the same slope.

    The slope at $\displaystyle x=-1$ for the first function is,
    $\displaystyle y'=2ax$ at $\displaystyle x=-1$ is $\displaystyle -2a$.
    The slope for the second is,
    $\displaystyle y'=4x+2$ at $\displaystyle x=-1$ is $\displaystyle -2$
    Thus, $\displaystyle -2=-2a$ thus, $\displaystyle a=1$
    All we need is to find "b" in,
    $\displaystyle y=x^2+b$ since (-1,0) is on the curve we have,
    $\displaystyle 0=(-1)^2+b$ thus, $\displaystyle b=-1$
    Thus,
    $\displaystyle (a,b,c)=(1,-1,2)$
    Which is not true.
    Thus, either you erred in the formulation of your question or there is no such numbers.
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  3. #3
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    Hello, jayson81!

    I got the same result as ThePerfectHacker, but I find that the solution works . . .

    The curves $\displaystyle y = ax^2 +b$ and $\displaystyle y = 2x^2 +cx$ have a common tangent at the point $\displaystyle (-1,0)$.
    Find $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$.
    Since $\displaystyle (-1,0)$ is on both curves, we have:
    . . $\displaystyle a(-1)^2 + b \,=\,0\quad\Rightarrow\quad a + b \,=\,0$
    . . $\displaystyle 2(-1)^2 + c(-1)\,=\,0\quad\Rightarrow\quad c = 2$

    Since the slopes are equal at $\displaystyle (-1,0)$, we have:
    . . $\displaystyle 2ax\,=\,4x + c\quad\Rightarrow\quad -2a\,=\,-4 + c$
    Since $\displaystyle c = 2$, we get: $\displaystyle a = 1$ . . . and hence: $\displaystyle b = -1$

    Therefore: .$\displaystyle (a,b,c)\,=\,(1,-1,2)$


    The parabolas are: .$\displaystyle y\,=\,x^2 - 1$ and $\displaystyle y\,=\,2x^2 + 2x$
    They share the common point $\displaystyle (-1,0)$ where they both have slope $\displaystyle -2.$
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