Math Extra Credit

Math Extra Credit - Lucas Vascocu Math 241 Extra Credit...

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Lucas Vascocu February 14, 2008 Math 241 Extra Credit Let (a,b) be such a point on the ellipse. It satisfies the equation of the ellipse:  a + 4b = 36. Now to obtain the derivative of the ellipse: x 2  + 4y 2  = 36  >>>  2x + 8y(dy/dx) = 0  >>>  8y(dy/dx) = -2x   >>>  dy/dx = (-2x/8y)  >>>  dy/dx =  -x/4y At such a point (a,b) the slope of the tangent line is –a/4b. Thus using the slope form of the line:  (y-b) = (-a/4b)(x-a)  Multiply by 4b to simplify the equation: 4by - 4b 2  = -ax + a 2   >>>  4by + ax = a 2  + 4b 2   >>>  4by + 4ax = 36 4by + ax = 36 is now the tangent line of the ellipse at the point (a,b). Now, the tangent line must contain the point (12,3).  Thus, 12a + 12b = 36 while a + b = 3. Obtaining this information, we get two values for b. (3-b) 2  + 4b 2  = 36  >>>  b 2  - 6b + 9 + 4b 2  = 36  >>>  5b 2
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