Solution

The correct answer is A-II, B-IV, C-I, D-III.

Key Points

• A. One-sheeted hyperboloid → II
  • Standard form: x²/a² + y²/b² − z²/c² = 1.
  • Two positive terms and one negative term.
• B. Two-sheeted hyperboloid → IV
  • Standard form: z²/c² − x²/a² − y²/b² = 1.
  • One positive term and two negative terms.
• C. Elliptic paraboloid → I
  • Standard form: x²/a² + y²/b² = cz.
  • Both squared terms positive.
• D. Hyperbolic paraboloid → III
  • Standard form: x²/a² − y²/b² = cz.
  • One positive and one negative squared term.

Additional Information

• Quick Identification Rule:
  • Two + and one − ⇒ One-sheet hyperboloid.
  • One + and two − ⇒ Two-sheet hyperboloid.
  • All + on one side with linear variable ⇒ Elliptic paraboloid.
  • Mixed signs with linear variable ⇒ Hyperbolic paraboloid.