If x=rsinθcos∝ , y=rsinθ sin∝ and z=rcosθ , then

Correct option is B

Given:

​$$x = r \sin \theta \cos \alpha, \quad y = r \sin \theta \sin \alpha, \quad z = r \cos \theta$$​

Solution:

1. Calculate $$x^2, y^2, z^2$$​ :

​$$x^2 = \left( r \sin \theta \cos \alpha \right)^2 = r^2 \sin^2 \theta \cos^2 \alpha y^2 = \left( r \sin \theta \sin \alpha \right)^2 = r^2 \sin^2 \theta \sin^2 \alpha z^2 = \left( r \cos \theta \right)^2 = r^2 \cos^2 \theta$$​

2. Add $$x^2$$​ and $$y^2$$​ :

​$$x^2 + y^2 = r^2 \sin^2 \theta \cos^2 \alpha + r^2 \sin^2 \theta \sin^2 \alpha Factor out r^2 \sin^2 \theta : x^2 + y^2 = r^2 \sin^2 \theta \left( \cos^2 \alpha + \sin^2 \alpha \right) Using \cos^2 \alpha + \sin^2 \alpha = 1 : x^2 + y^2 = r^2 \sin^2 \theta$$​

3. Calculate $$x^2 + y^2 - z^2$$​ :

Substitute $$x^2 + y^2 = r^2 \sin^2 \theta and z^2 = r^2 \cos^2 \theta : x^2 + y^2 - z^2 = r^2 \sin^2 \theta - r^2 \cos^2 \theta Factor out r^2 : x^2 + y^2 - z^2 = r^2 \left( \sin^2 \theta - \cos^2 \theta \right) Using \sin^2 \theta + \cos^2 \theta = 1 , the above simplifies to: x^2 + y^2 - z^2 = r^2$$​

Final Answer:

​$$\mathbf{B. \; x^2 + y^2 - z^2 = r^2}$$​