Three Dimensional Geometry Question 361
Question: If the center of the sphere $ ax^{2}+by^{2}+cz^{2}-2x+4y+2z-3=0 $ is $ (1/2,-1,-1/2) $ , what is the value of b ?
Options:
A) 1
B) -1
C) 2
D) -2
Show Answer
Answer:
Correct Answer: C
Solution:
[c] The given equation of sphere is $ ax^{2}+by^{2}+cz^{2}-2x+4y+2z-3=0 $ This equation represents a equation of sphere, if coefficient of $ x^{2},y^{2} $ and $ z^{2} $ is same. i.e., a =b =c
$ \therefore $ Equation of sphere can be re-written as $ bx^{2}+by^{2}+bz^{2}-2x+4y+2z-3=0 $
$ \Rightarrow x^{2}+y^{2}+Z^{2}-\frac{2x}{b}+\frac{4y}{b}+\frac{2z}{b}-\frac{3}{b}=0 $ The centre of this sphere is $ ( \frac{1}{b},\frac{-2}{b},\frac{-1}{b} ) $ Given that the centre of sphere is $ ( \frac{1}{2},-1,-\frac{1}{b} ) $ $ \frac{1}{b}=\frac{1}{2}\Rightarrow b=2 $
BETA
