How do you use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces x2+y2=25 and z=3x2?

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umkhululiueyj · Accepted Answer

Step 1Use the following parametrization for the curve s generated by the intersection:s(t)=(x(t),y(t),z(t)),t∈[0,2π)x=5cos⁡(t)y=5sin⁡(t)z=75cos2(t)Note that s(t):R→R3 is a vector valued function of a real variable.To reach this result, consider the curves that these equations define on certain planes.Step 2The equation x2+y2=25 defines a circle of radius 5 centered on the z-axis on the planes z=c1, where c1∈R is any constant.The equation z=3x2 defines a parabola on any plane y=c2, where c2∈R is another constant.The surfaces are, therefore, those obtained by translating the circle along the z-axis and the parabola along the y-axis.To obtain a parametrization for the intersection curve s, we must find equations for x, y and z as functions of t that obey both equations given in the problem.Step 3Consider the standard parametrization for a circle C of radius r (it&#039;s easy to see that this parametrization fulfils the condition x2+y2=r2):C(t)=(rcos⁡(t),rsin⁡(t)),t∈[0,2π)Checking the first equation, we get r=5 andx=5cos⁡(t)y=5sin⁡(t)Now, we already have an expression for x(t). So, in order to obey the second condition, we make:z=3x3=3(5cos⁡(t))2=75cos2(t)And we have the parametrization s(t)=(x(t),y(t),z(t)),t∈[0,2π)

How do you use cos(t) and sin(t), with

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