Question

The equation of the circle passing through \[\left( {2,0} \right)\] and \[\left( {0,4} \right)\] and having the minimum radius is
A \[{x^2} + {y^2} = 20\]
B \[{x^2} + {y^2} - 2x - 4y = 0\]
C \[\left( {{x^2} + {y^2} - 4} \right) + \lambda \left( {{x^2} + {y^2} - 16} \right) = 0\]
D N.O.T.

Answer 1

Hint: In this problem, first we need to find the coordinate of the centre of the circle. Now, substitute the coordinate of the centre into the equation of the circle. Next, pass the equation of the circle through any of the given points to obtain the radius of the circle.

Complete step-by-step answer:
The equation of the circle having centre \[\left( {h,k} \right)\] and radius \[r\] is as follows:
\[{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( 1 \right)\]
Consider, the line joining the points \[\left( {2,0} \right)\] and \[\left( {0,4} \right)\] be the diameter of the circle. Now, the midpoint of the line joining the points \[\left( {2,0} \right)\] and \[\left( {0,4} \right)\] represents the centre \[\left( {h,k} \right)\] of the circle.
The midpoint of the line joining the points \[\left( {2,0} \right)\] and \[\left( {0,4} \right)\] is calculated as follows:
\[\begin{gathered}
  h = \dfrac{{2 + 0}}{2} = 1 \\
  k = \dfrac{{0 + 4}}{2} = 2 \\
\end{gathered}\]
Thus, the centre \[\left( {h,k} \right)\] of the circle is\[\left( {1,2} \right)\].
Now, substitute 1 for \[h\] and 2 for \[k\] in equation (1).
\[{\left( {x - 1} \right)^2} + {\left( {y - 2} \right)^2} = {r^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( 2 \right)\]
Since, the circle passes through the point \[\left( {2,0} \right)\], substitute 2 for \[x\] and 0 for \[y\] in equation (2) to obtain the radius of the circle.
\[\begin{gathered}
  \,\,\,\,\,\,{\left( {2 - 1} \right)^2} + {\left( {0 - 2} \right)^2} = {r^2} \\
   \Rightarrow {\left( 1 \right)^2} + {\left( { - 2} \right)^2} = {r^2} \\
   \Rightarrow 1 + 4 = {r^2} \\
   \Rightarrow 5 = {r^2} \\
\end{gathered}\]
Thus, the radius of the circle having minimum radius is shown below.
\[\begin{gathered}
  \,\,\,\,\,\,\,{\left( {x - 1} \right)^2} + {\left( {y - 2} \right)^2} = 5 \\
   \Rightarrow {x^2} + 1 - 2x + {y^2} + 4 - 4y = 5 \\
   \Rightarrow {x^2} + {y^2} - 2x - 4y + 5 - 5 = 0 \\
   \Rightarrow {x^2} + {y^2} - 2x - 4y = 0 \\
\end{gathered}\]

Thus, the option (B) is the correct answer.

Note: The line joining the given points represents the diameter of the circle. The midpoint of the diameter represents the coordinate of the centre of the circle.