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Unformatted text preview: 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 2 + 4b 2 = 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....
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