Pre-Calculus

Write the equations in cylindrical coordinates. (a) 3x2−2x+3y2+z2=1 Write your answer for part (a). Press Space or Enter to edit this math answer. Correct: Your answer is correct. (b) z=6x2−6y2

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Recall cylindrical coordinate substitutions

In cylindrical coordinates, we replace x and y using these formulas. The variable z stays the same.
2

Substitute x and y into the equation

We replace x with and y with directly into .
3

Expand the squared terms

We square each term: and .
4

Factor out the common factor

Both terms share the factor , so we factor it out to simplify the expression.
5

Apply the double-angle identity

The identity lets us write this in a clean, compact form.

Understanding this problem

Learning Insight

Cylindrical coordinates work by describing a point's horizontal position with a radius r and angle θ, just like polar coordinates, and keeping z for height. When you substitute and , trig identities like the double-angle formula often appear naturally and simplify the result beautifully.

Quick Tip

Whenever you see in a problem, think double-angle right away: after substitution it always becomes . Similarly, is the other key pattern to memorize.

Common Mistake

Students often forget to factor correctly and leave the answer as without applying the double-angle identity, missing the fully simplified cylindrical form.