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Math Help - Help on Parametric Curves please

  1. #1
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    Help on Parametric Curves please

    I have no idea how to do this math problem.

    x = (4(t^3) + 1)/4
    y = 3(t^5) +1
    Find all points (a,b) on this curve such that the tangent line to the curve at (a, b) passes through the point (1/5, 1).

    Help?!
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  2. #2
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    Quote Originally Posted by raven17 View Post
    I have no idea how to do this math problem.

    x = (4(t^3) + 1)/4
    y = 3(t^5) +1
    Find all points (a,b) on this curve such that the tangent line to the curve at (a, b) passes through the point (1/5, 1).

    Help?!
    First we need some info on the point (a,b) putting this into the equation for x gives

    \displaystyle a=t^3+\frac{1}{4} \iff t=\left(a-\frac{1}{4}\right)^{\frac{1}{3}}

    Now putting this into the equation for y gives

    \displaystyle y=3\left(a-\frac{1}{4}\right)^{\frac{5}{3}}+1

    So the ordered pair \displaystyle \left(a,3\left(a-\frac{1}{4}\right)^{\frac{5}{3}}+1\right) must lie on the graph.

    Now we can use this generic point to find the slope passing through the point \displaystyle \left( \frac{1}{5},1\right) this gives

    \displaystyle m=\frac{3\left(a-\frac{1}{4}\right)^{\frac{5}{3}}+1-1}{a-\frac{1}{5}}= \frac{3\left(a-\frac{1}{4}\right)^{\frac{5}{3}}}{a-\frac{1}{5}}

    Now we also know the slope at this point must be

    \displaystyle \frac{dy}{dx}\bigg|_{x=a} so...

    Can you finish from here?
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  3. #3
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    Quote Originally Posted by raven17 View Post
    I have no idea how to do this math problem.

    x = (4(t^3) + 1)/4
    y = 3(t^5) +1
    Find all points (a,b) on this curve such that the tangent line to the curve at (a, b) passes through the point (1/5, 1).

    Help?!

    \displaystyle x = {{4\,t^3 + 1}\over4}\ \ \to\ \ x = t^3 + {1\over4}

    y = 3\,t^5 +1

    \displaystyle  {{dy}\over{dx}}={{dy/dt}\over{dx/dt}}={{15t^4}\over{3t^2}} =5t^2

    Slope from \displaystyle ({1\over5},\ 1) to (x,\ y) must equal 5t^2.

    \displaystyle {{3\,t^5 +1-1}\over{t^3 + {1\over4}-{1\over5}}} =5t^2.

    Solve for \displaystyle t (There are 2 solutions.) and use \displaystyle t to find (x,\ y).

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  4. #4
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    Thanks soo much! That really helped.
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  5. #5
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    Thank you too! I understood how to finish up the problem thanks to you.
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