Final answer:
The Cartesian equation for the curve r=5 csc σ is x²=24y², representing a circle centered at the origin with a radius √24.
Step-by-step explanation:
The student is asking for the Cartesian equation of the curve r=5 csc σ and the identification of the type of curve it represents. In polar coordinates, r is the radius, and σ (sigma) is the angle. The function csc(σ) = 1/sin(σ), so this equation can also be written as r=5/sin(σ). To convert this to Cartesian coordinates, we note that r = √(x²+y²) and sin(σ) = y/r. Substituting these into the polar equation, we get r=5/(y/√(x²+y²)). Simplifying, r√(x²+y²)=5y. Since r=√(x²+y²), after squaring both sides we obtain x²+y²=(5y)², which is equivalent to x²+y²=25y². This simplifies to x²=24y², which recognizes the curve as a circle centered at the origin with a radius √of 24.