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**Hint**: Orthogonal circles cut one another at right angles. Using Pythagoras theorem, two circle of radii r1, r2 whose center are at distance d apart are orthogonal if \[r_1^2 + r_2^2 = {d^2}\] and given by the equation \[2gg' + 2ff' = c + c'\].

The angle of intersection of two overlapping circles is defined as the angle between their tangents at their intersection points. Where if the angle is \[{180^ \circ }\] it is known as tangent and the angle is \[{90^ \circ }\]then it is orthogonal.

**:**

__Complete step-by-step answer__Given the equation of one circle is \[{x^2} + {y^2} = 9\] whose center is (0, 0)

Given the radical axis of two circles\[x = 1\], therefore the line joining centers should be perpendicular to\[x = 1\]

The center of another circle should lie on\[y = 0\],

Let the center of another circle be \[\left( {a,0} \right)\]and the radius \[r\],

So the equation of other circle is \[{\left( {x - a} \right)^2} + {y^2} = {r^2} - - - - (i)\]

Since the given two circles are orthogonal we can write \[{a^2} = {r^2} + 9\]

We know the center of the two circle are at \[\left( {0,0} \right)\] and also their radius being 3 and r respectively

Since the lengths of the tangents from radial axis are equal, hence we get

\[\Rightarrow {\left( {1 - a} \right)^2} + 0 - {r^2} = 1 - 9\]

This can be written as:

\[

\Rightarrow {\left( {1 - a} \right)^2} + 0 - {r^2} = 1 - 9 \\

\Rightarrow 1 - 2a + {a^2} - {r^2} = - 8 \\

\Rightarrow {a^2} - {r^2} = 2a - 9 - - (i) \\

\]

We know \[{a^2} = {r^2} + 9\] since the circles are orthogonal

\[\Rightarrow {a^2} - {r^2} = 9 - - (ii)\]

Hence by solving equation (i) and (ii) we can write

\[

\Rightarrow 2a - 9 = 9 \\

\Rightarrow 2a = 18 \\

\Rightarrow a = 9 \\

\]

Also

\[

{a^2} - {r^2} = 9 \\

\Rightarrow {\left( 9 \right)^2} - {r^2} = 9 \\

\Rightarrow {r^2} = 81 - 9 \\

\Rightarrow {r^2} = 72 \\

\]

Substituting the values of $ a = 9 $ and $ {r^2} = 72 $ in equation (i) as:

\[

\Rightarrow {\left( {x - a} \right)^2} + {y^2} = {r^2} \\

\Rightarrow {\left( {x - 9} \right)^2} + {y^2} = 72 \\

\Rightarrow {x^2} + 81 - 18x + {y^2} - 72 = 0 \\

\Rightarrow {x^2} + {y^2} - 18x + 9 = 0 \\

\]

Hence, the equation of the required circle is given as \[{x^2} + {y^2} - 18x + 9 = 0\].

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

**Note**: Students must not get confused with the two equations of the circle as the coordinates of the center of both the circles are different. Moreover, the radius is varying by a considerable amount. Always try to stick to the fundamental standard equation of the circle i.e., $ {\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {r^2} $ .

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