Evolutes
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Evolutes Exercise 1A
(1)Find parametric equation for the evolute of the hyperbola [Equation goes here - download the original to see it.] (2)Let C be the curve [Equation goes here - download the original to see it.] Find the parametric equation of the evolute of the curve C. (3) Find the parametric equation of the evolute of the curve [Equation goes here - download the original to see it.] (4) Show that the equation of the normal of the circle [Equation goes here - download the original to see it.] can be written in the form [Equation goes here - download the original to see it.] Find the equation of the evolute of this circle. (5) Prove that the evaluate of the curve Equation goes here - download the original to see it.] has parametric equation [Equation goes here - download the original to see it.] (6) Prove that evaluate of the logarithmic spiral [Equation goes here - download the original to see it.] is also a logarithmic spiral with the same pole. (7) Find the parametric equations for the evolute of the curve [Equation goes here - download the original to see it.] (8) Find the parametric equations of the evolute of the curve [Equation goes here - download the original to see it.] in two ways. Solutions Exercise 1A (1) The parametric equation of the hyperbola [Equation goes here - download the original to see it.] is [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The gradient of the normal is [Equation goes here - download the original to see it.] and the equation of the normal is [Equation goes here - download the original to see it.] Substituting this in [Equation goes here - download the original to see it.] gives [Equation goes here - download the original to see it.] Therefore, the parametric equation of the evolute of the hyperbola is [Equation goes here - download the original to see it.] (2)[Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Therefore, the gradient of the normal is [Equation goes here - download the original to see it.] The equation of the normal is [Equation goes here - download the original to see it.] (1) [Equation goes here - download the original to see it.] From this [Equation goes here - download the original to see it.] Substituting this into (1), we get [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Therefore, the parametric equation of the evaluate is [Equation goes here - download the original to see it.]
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Contents of
Evolutes
1 Evolutes Exercise 1A
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