Question

The equation of the circle passing through $\left( {4,5} \right)$ and having the center $\left( {2,2} \right)$ is
A. ${x^2} + {y^2} + 4x + 4y - 5 = 0$
B. ${x^2} + {y^2} - 4x - 4y - 5 = 0$
C. ${x^2} + {y^2} - 4x = 13$
D. ${x^2} + {y^2} - 4x - 4y - 5 = 0$

Answer 1

Hint: In this question, we are given the coordinates of the center and the point through which the circle is passing. Now, use the general equation of the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$whose center is $\left( { - g, - f} \right)$. Here, calculate the value of $g$and $f$ using $k = - g$, $h = - f$ where $\left( {h,k} \right)$are $\left( {2,2} \right)$. Now put the value of $x$ , $y$ in the required equation of the circle to get the value of $c$. Lastly, put the value of $c$ in the equation and you’ll get the equation of the circle.

Formula Used:
General equation of circle, ${x^2} + {y^2} + 2gx + 2fy + c = 0$

Complete step by step solution:
Given that,
Center of the circle is $\left( {2,2} \right)$ i.e., $\left( {h,k} \right)$

Image: Circle
The general form of the equation of circle is ${x^2} + {y^2} + 2gx + 2fy + c = 0$whose center is $\left( { - g, - f} \right)$
Therefore, $k = - g$, $h = - f$
It implies that, $g = - 2$, $f = - 2$
Now, the equation of the circle is ${x^2} + {y^2} - 4x - 4y + c = 0 - - - - - - \left( 1 \right)$
Also, the equation is passing through the point $\left( {4,5} \right)$, which means points will satisfy the above equation.
Put $x = 4$ and $y = 5$ in equation (1)
${\left( 4 \right)^2} + {\left( 5 \right)^2} - 4\left( 4 \right) - 4\left( 5 \right) + c = 0$
$16 + 25 - 16 - 20 + c = 0$
$c = - 5$
Equation (1) will be,
Equation of the circle is ${x^2} + {y^2} - 4x - 4y - 5 = 0$

Option ‘B’ is correct

Note: To solve such a question, first try to make the figure and understand the question properly. One should always remember each and every equation of the circle (General and standard both). Also, the formulas to find the radius and the distance between any point and line.