Question

The equation $a{x}^2+a{y}^2+a{z}^2+2ux+2vy$ $+2wz+d=0,a\ne 0$ represent a sphere if

• A. $u^2+v^2+w^2+ad\le 0$
• B. $u^2+v^2+w^2+ad\ge 0$
• C. $u^2+v^2+w^2-ad\le 0$
• D. $u^2+v^2+w^2-ad\ge 0$

Answer 1

The correct option is D $u^2+v^2+w^2-ad\ge 0$

Given $a{x}^2+a{y}^2+a{z}^2+2ux+2vy+2wz+d=0$

$\Rightarrow x^2+y^2+z^2+2\left(\frac{u}{a}\right)x+2\left(\frac{v}{a}\right)y+2\left(\frac{w}{a}\right)z+\frac{d}{a}=0$

Add and substract ${\left(\frac{u}{a}\right)}^2+{\left(\frac{v}{a}\right)}^2+{\left(\frac{w}{a}\right)}^2$ on both sides.

$\Rightarrow x^2+y^2+z^2+2\left(\frac{u}{a}\right)x+2\left(\frac{v}{a}\right)y+2\left(\frac{w}{a}\right)z+{\left(\frac{u}{a}\right)}^2+{\left(\frac{v}{a}\right)}^2+{\left(\frac{w}{a}\right)}^2={\left(\frac{u}{a}\right)}^2+{\left(\frac{v}{a}\right)}^2+{\left(\frac{w}{a}\right)}^2-\frac{d}{a}$

${(x+\frac{u}{a})}^2+{(y+\frac{v}{a})}^2+{(z+\frac{w}{a})}^2={\left(\sqrt{\frac{u^2+v^2+w^2-ad}{a}}\right)}^2$

$r=\frac{\sqrt{u^2+v^2+w^2-ad}}{a}$

$\Rightarrow u^2+v^2+w^2-ad\ge 0$