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Mathematics-Online problems:

Interactive Problem 54: Tangent Line to a Parabola and Minimum Area

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
The tangent line $ g$ to the parabola

$\displaystyle y = x(1-x) $

intersects the positive $ x-\!$axis at point $ A$ and the positive $ y-\!$axis at point $ B$, respectively.


\includegraphics[width=.6\linewidth]{TdM_13_A1_bild}


Find the equation of the tangent line $ g$ as a function of the $ x-\!$coordinate $ t > 1/2$ of the point of tangency $ P$.
For which point $ P_{\min}$ is the area $ F$ of the triangle $ \bigtriangleup (O, A, B)$ minimal, and what is the size of the minimum area $ F_{\min}$?


Answer:

$ g$ for $ t=3/4: y = $ $ x$ +  
$ P_{\min} = $ $ \big($ , $ \big)$  
$ F_{\min} = $  

(The results should be correct to four decimal places.)


   

(From: Mathematics Contest 2013)
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  automatically generated: 2/ 6/2018