question_answer

The equation of the circle which touches the axes at a distance 5 from the origin is \[{{y}^{2}}+{{x}^{2}}-2ax-2ay+{{a}^{2}}=0.\] what is the value of\[\alpha \]?

A) 4

B) 5

C) 6

D) 7

Correct Answer: B

Solution :

[b] Coordinates of the centre of given circle \[=(\alpha ,\,\,\alpha )\] and radius \[=\sqrt{{{(\alpha )}^{2}}+{{(\alpha )}^{2}}-{{\alpha }^{2}}}=\sqrt{{{\alpha }^{2}}}=\alpha \] \[\therefore {{(\alpha -5)}^{2}}+{{(\alpha )}^{2}}={{(\alpha )}^{2}}\] \[\Rightarrow {{\alpha }^{2}}+25-10\alpha =0\Rightarrow {{(\alpha -5)}^{2}}=0\Rightarrow \alpha =5\] Then, other root will always real.