Question

State the following statement is True or False

The co-ordinates of the ends of a diameter of a sphere are $(2,-3,4)$ and $(-5,6,7)$ and equation is $x^2+y^2+z^2+3x-3y+3z-56=0$

• A. True
• B. False

Answer 1

The correct option is B False

The end points of the diameter of a sphere are given by $(2,-3,4)$ and $(-5,6,7)$

The midpoint of these points would be the center of the sphere.

It turns out to be $(\frac{2-5}{2},\frac{-3+6}{2},\frac{4+7}{2})=(-1.5,1.5,5.5)$

Also, distance between those points is $\sqrt{(2-(-5){)}^2+(-3-6{)}^2+(4-7{)}^2}=\sqrt{49+81+9}=\sqrt{139}$

Thus, the radius is given by $\frac{139}{2}$

Now, the equation of the sphere can be written as $(x+1.5{)}^2+(y-1.5{)}^2+(z-5.5{)}^2={\left(\frac{\sqrt{139}}{2}\right)}^2$

$\therefore x^2+3x+\frac{9}{4}+y^2-3y+\frac{9}{4}+z^2-11z+\frac{121}{4}=\frac{139}{4}$

$\Rightarrow x^2+y^2+z^2+3x-3y-11z=0$ is the required equation.