question_answer

The centre of a circle is \[(2a,\,a-7)\]. If the circle passes through the point \[(1,-9)\] and has diameter \[10\sqrt{2}\]c units, then the value of a is:

A) \[9\]

B) \[-\sqrt{3}\]

C) \[\sqrt{3}\]        

D) \[\pm \sqrt{3}\]

Correct Answer: D

Solution :

[d] Let \[O(2a,\,\,a-7)\]be the centre and \[A(1,-9)\]be any point through which circle passes.

\[\therefore \,\,\,\,\,\,\,\,\,2(OA)=10\sqrt{2}\]

\[[Diameter=2\times radius]\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,OA=5\sqrt{2}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,O{{A}^{2}}={{(5\sqrt{2})}^{2}}=50\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,{{(2a-1)}^{2}}+{{(a-7+9)}^{2}}=50\]

\[\Rightarrow \,\,4{{a}^{2}}+1-4a+{{a}^{2}}+4+4a=50\,\,\,\Rightarrow \,\,5{{a}^{2}}=50-5=45\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{a}^{2}}=9\,\,\,\,\,\,\Rightarrow a=\pm 3\]