Question

Consider a sphere passing through the origin and the points $(2,1,-1),(1,5,-4),(-2,4,-6)$.

Consider the following statements:
$1$. The sphere passes through the point $(0,4,0)$.
$2$. The point $(1,1,1)$ is at a distance of $5$ unit from the centre of the sphere.
Which of the statements is$/$are correct?

• A. $1$ only
• B. $2$ only
• C. Both $1$ and $2$
• D. Neither $1$ nor $2$

Answer 1

The correct option is A $1$ only

1. Equation of sphere is

$x^2+y^2+z^2+2x-4y+6z=0$

Put the value $(0,4,0)$, we get

$0+16+0+0-16+0=0$

So the sphere passes through the point $(0,4,0)$

Hence Statement 1 is correct

2. Distance between the point $(1,1,1)$ and centre of sphere $(-1,2,-3)$

$⟹\sqrt{(1+1{)}^2+(1-2{)}^2+(1+3{)}^2}$

$⟹\sqrt{4+1+16}$

$⟹\sqrt{21}$ is not equal to $5$

Hence Statement 2 is incorrect