Question

The circles $x^{2}+y^{2}-10x+16=0$ and $x^{2}+y^{2}=a^{2}$ intersect at two distinct points, if

• A. $a <\, 2$
• B. $2 <\, a <\, 8$
• C. $a >\, 8$
• D. $a = 2$

Answer 1

Answer: (B)

Here, centres and radii of given circles are

$C_{1}\left(5, 0\right), r_{1}=\sqrt{25+0-16}=3$

and $C_{2} \left(0, 0\right), r_{2}=a$

Now, $C_{1}C_{2}=\sqrt{\left(5-0\right)^{2}+0^{2}}=5$

Since, two circles intersect at two distinct points

$\therefore \left|r_{1}-r_{2}\right| <\, C_{1}C_{2}<\, r_{1}+r_{2}$

$\Rightarrow \left|a-3\right|<\,5<\,a+3$

$\Rightarrow \left|a-3\right|<\,5$ and $5<\, a+3$

$\Rightarrow -5<\,a-3 <\,5$ and $a>\, 2 $

$\Rightarrow -2 <\, a<\,8$ and $a>\,2$

$\Rightarrow 2 <\, a <\,8$