Question

Parametrize the intersection of the surfaces x² + y² = z² , 3y = z² using t = z as parameter. (Use symbolic notation and fractions where needed.

Answer 1

To parametrize the intersection of the surfaces x² + y² = z² and 3y = z² using t = z as the parameter, express x, y, and z in terms of t.

Step-by-step explanation:

To parametrize the intersection of the surfaces x² + y² = z² and 3y = z² using t = z as the parameter, we need to express x, y, and z in terms of t. Let's solve both equations for z:

From x² + y² = z², we have z = sqrt(x² + y²).

From 3y = z², we have z = sqrt(3y).

Equating the two expressions for z, we get sqrt(x² + y²) = sqrt(3y). Squaring both sides, we have x² + y² = 3y. Rearranging the equation, we get y = (x²)/(3-x).

Finally, we can express x, y, and z in terms of t:

x = t, y = (t²)/(3-t), and z = sqrt(x² + y²) = sqrt(t² + (t²)/(3-t)).