Answer

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**Hint:**We here have been given the equation of the sphere as ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4x+6y-8z-7=0$ and we have to find its diameter. For this, we will equate this equation with the general equation of the sphere, i.e. ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2fx+2gy+2hz+c=0$ and get the values of f, g, h and c from the given equation. Then we will use the formula for the radius of a sphere given as $r=\sqrt{{{f}^{2}}+{{g}^{2}}+{{h}^{2}}-c}$ and hence obtain the radius of the sphere. Then we will use the fact that the diameter of a sphere is twice its radius and hence find the value of the diameter.

**Complete step by step answer:**

Here, we have been given the equation of a sphere as ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4x+6y-8z-7=0$ and we have to find its diameter.

Now, we can see that this equation of the sphere is in the form of ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2fx+2gy+2hz+c=0$ which is the general equation of a sphere.

We also know that the radius of a sphere with the equation ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2fx+2gy+2hz+c=0$ is given as:

$r=\sqrt{{{f}^{2}}+{{g}^{2}}+{{h}^{2}}-c}$

Where r is the radius of the sphere.

Now, if we equate the given equation of the sphere, i.e. ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4x+6y-8z-7=0$ with the general equation of the sphere ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2fx+2gy+2hz+c=0$, we will get:

2f=-4

2g=6

2h=-8

c=-7

Hence, on solving these, we get the value of f, g and h as:

$\begin{align}

& f=\dfrac{-4}{2}=-2 \\

& g=\dfrac{6}{2}=3 \\

& h=\dfrac{-8}{2}=-4 \\

\end{align}$

Now, if we put these values of f, g, h and c in the formula for radius, we will get the radius of the required sphere.

Thus, putting the values of f, g, h and c in the formula for radius, we get:

$\begin{align}

& r=\sqrt{{{f}^{2}}+{{g}^{2}}+{{h}^{2}}-c} \\

& \Rightarrow r=\sqrt{{{\left( -2 \right)}^{2}}+{{\left( 3 \right)}^{2}}+{{\left( -4 \right)}^{2}}-\left( -7 \right)} \\

\end{align}$

Now, solving this, we get the radius as:

$\begin{align}

& r=\sqrt{{{\left( -2 \right)}^{2}}+{{\left( 3 \right)}^{2}}+{{\left( -4 \right)}^{2}}-\left( -7 \right)} \\

& \Rightarrow r=\sqrt{4+9+16+7} \\

& \Rightarrow r=\sqrt{36} \\

& \Rightarrow r=6 \\

\end{align}$

Thus, the radius of the given sphere is 6 units. But here, we need to find the diameter.

Now, we know that the diameter of any sphere is twice its radius.

Thus, we can say that:

$diameter=2\left( radius \right)$

Putting the value of radius in this, we get:

$\begin{align}

& diameter=2\left( radius \right) \\

& \Rightarrow diameter=2\left( 6 \right) \\

& \therefore diameter=12units \\

\end{align}$

Thus, the diameter of the given sphere is 12 units.

**So, the correct answer is “Option D”.**

**Note:**We can also do this question by the following method:

Now, we have been given the equation of the sphere as ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4x+6y-8z-7=0$.

Now, we know that the general equation of a sphere is given as:

${{\left( x-\alpha \right)}^{2}}+{{\left( y-\beta \right)}^{2}}+{{\left( z-\gamma \right)}^{2}}={{r}^{2}}$

Where $\left( \alpha ,\beta ,\gamma \right)$ is its centre and r is the radius.

Thus, we can change the given equation in this form and get our radius as:

$\begin{align}

& {{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4x+6y-8z-7=0 \\

& \Rightarrow \left( {{x}^{2}}-4x \right)+\left( {{y}^{2}}+6y \right)+\left( {{z}^{2}}-8y \right)-7=0 \\

\end{align}$

Adding 4, 9 and 16 on the LHS and the RHS, we get:

$\begin{align}

& \left( {{x}^{2}}-4x \right)+\left( {{y}^{2}}+6y \right)+\left( {{z}^{2}}-8y \right)-7=0 \\

& \Rightarrow \left( {{x}^{2}}-4x \right)+\left( {{y}^{2}}+6y \right)+\left( {{z}^{2}}-8y \right)-7+4+9+16=4+9+16 \\

& \Rightarrow \left( {{x}^{2}}-4x+4 \right)+\left( {{y}^{2}}+6y+9 \right)+\left( {{z}^{2}}-8y+16 \right)-7=29 \\

& \Rightarrow {{\left( x-2 \right)}^{2}}+{{\left( y+3 \right)}^{2}}+{{\left( z-4 \right)}^{2}}=29+7 \\

& \Rightarrow {{\left( x-2 \right)}^{2}}+{{\left( y+3 \right)}^{2}}+{{\left( z-4 \right)}^{2}}=36 \\

& \Rightarrow {{\left( x-2 \right)}^{2}}+{{\left( y+3 \right)}^{2}}+{{\left( z-4 \right)}^{2}}={{\left( 6 \right)}^{2}} \\

\end{align}$

Thus, the radius of the given sphere is 6.

Hence, the diameter is given as:

$\begin{align}

& diameter=2\left( radius \right) \\

& \Rightarrow diameter=2\left( 6 \right) \\

& \therefore diameter=12units \\

\end{align}$

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