Question

	LIST I (Name of the surface)		LIST II
A.	Ellipsoid		I. x²/a² + y²/b² + z²/c² = 1
B.	Hyperboloid of one sheet		II. x²/a² - y²/b² - z²/c² = 1
C.	Hyperboloid of two sheets		III. -x²/a² + y²/b² + z²/c² = 1
D.	Central conicoid		IV. x²/a² + y²/b² + z = 1

Choose the correct answer from the options given below:

• (A) - (I), (B) - (II), (C) - (III), (D) - (IV)
• (A) - (I), (B) - (III), (C) - (II), (D) - (IV)
• (A) - (IV), (B) - (II), (C) - (I), (D) - (III)
• (A) - (IV), (B) - (III), (C) - (II), (D) - (I)

Hint:

Each surface equation has a distinct form depending on the signs and coefficients of \( x^2 \), \( y^2 \), and \( z^2 \).

Answer 1

The Correct Option is D

The standard equations for the different surfaces are:
Ellipsoid: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \)
Hyperboloid of one sheet: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \)
Hyperboloid of two sheets: \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \)
Central conicoid: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + z = 1 \)