Consider the curve x^2/3+y^2/3=4 . Let F be the tangent line to this curve at the point (1, 3–√3) , and let C and D be the x- and y-intercepts of F .

Question

What is the length of the line segment CD ?

Answer 1

For the equation x² + y² = 12 (not that this is a circle of radius 2sqrt(3) centered at the origin), it is easier to differentiate explicitly than to solve for y and take the derivative :

2x + 2y dy/dx = 0 therefore, dy/dx = -x/y (which we new because x over and y up is the point which has a radius of y/x and the tangent must be the negative reciprocal of the perpendicular: -x/y)

The equation of the line can be written in slope form ( y - f(x*))/ (x - x*) = - x*/y* which just says that any point along the line must have a slope equal to the slope at the known point. (x*,y*) = (1, 3-sqrt(3))

You can plug 0 in for x and find the y intercept. Plug in 0 for y for the x intercept. The distance between them will just be the hypotenuse of the triangle between the x axis, y axis, and the line segment (Pyth. Thm)

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