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 $a>2$

$\Rightarrow 2<a<8$