# A circle of Radius $2$ unit lies in the First Quadrant and touches both the axes of coordinates, equation of circle with center at $(6,5)$ and touching the above circle externally is  $A -$ ${(x - 6)^2} + {(y - 5)^2} = 9$  $B -$ ${x^2} + {y^2} - 10x - 12y + 52 = 0$  $C -$ ${x^2} + {y^2} - 12x - 10y + 32 = 0$  $D -$ ${x^2} + {y^2} + 12x - 10y - 32 = 0$

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Hint: In order to solve numerical geometry we need to know each and every property along with all the formulae. For this sum we will have to use the property Sum of the radius=Distance between their centers. We have used this because it is given circle touch externally. Since this is a circle question we will use ${(x - h)^2} + {(y - k)^2} = {c^2}$ as the general equation

Let the centre of the given circle be $(h,k)$
Given that circle lies in the First Quadrant and touches both the axes of coordinates.
$\therefore h = k = 2$
The equation of the circle formed with Radius $2$ units and value of $h,k = 2$ is
${(x - 2)^2} + {(y - 2)^2} = {2^2}...............(1)$
Now, Let the radius of the required circle be $R$ .
$\therefore$ Equation of Circle with coordinates $(6,5)$ and radius as $R$ is
${(x - 6)^2} + {(y - 5)^2} = {R^2}...............(2)$
Given that Circle $1$ and Circle $2$ touch each other externally, We have to use the property
Sum of the radius=Distance between their Centre
The distance between the centre can be calculated by using the distance formula which is given by
$\sqrt {{{({x_1} - a)}^2} + {{({y_1} - b)}^2}}$ , where $({x_1},{y_1})\& (a,b)$ are coordinates of $2$ points between which we have to find the distance.
$\therefore$ $\Rightarrow R + 2 = \sqrt {{{(6 - 2)}^2} + {{(5 - 2)}^2}}$
$\Rightarrow R + 2 = \sqrt {{4^2} + {3^2}}$
$\Rightarrow R + 2 = \sqrt {25}$
Since Radius Is always Positive we will take only positive values of the root.
$\Rightarrow R + 2 = 5$
$\Rightarrow R = 3$
Substituting value of $R$ in $Equation2$ ,we get the Equation of the Required Circle is
$\Rightarrow {(x - 6)^2} + {(y - 5)^2} = {3^2}$
$\Rightarrow {(x - 6)^2} + {(y - 5)^2} = 9$
Thus answer to the given solution is Option $A$ i.e. ${(x - 6)^2} + {(y - 5)^2} = 9$
So, the correct answer is “Option A”.

Note: It is advisable to learn distance formulae and general equations for figures like Parabola, Hyperbola, Ellipse, Circle. It would help to solve problems where there are more than $1$ geometric figures involved for finding the equation for distance between the two figures.