(a) Note that (-1,0) lies on the curve x? + y² = 1. For t e R, denote by le the line with slope t that passes through (-1,0). This line intersects the curve x² + y? = 1 at (-1,0) as well as at one additional point (x(t), y(t)). In fact, since every point on the curve x² + y? = 1 other than (-1,0) is realized as the nontrivial intersection point of the line li for precisely one value of t, we obtain the parametrization E R² 2? + y? = 1} = {(#(t), y(t)) | t e R}U{(-1,0)}. Compute the values of x(t) and y(t) to obtain a parametrization of the real points lying on the curve x² + y² = 1.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.