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

The correct option is B $2<a<8$

Here, centres and radii of given circles are
$C_1(5,0),r_1=\sqrt{25+0-16}=3$
and $C_2(0,0),r_2=a$
Now, $C_1{C}_2=\sqrt{{(5-0)}^2+0^2}=5$
Since, two circles intersect at two distinct points.
$\therefore |r_1-r_2|<C_1{C}_2<r_1+r_2$
$\Rightarrow |a-3|<5<a+3$
$\Rightarrow |a-3|<5$ and $5<a+3$
$\Rightarrow -5<a-3<5$ and $a>2$
$\Rightarrow -2<a<8$ and $a>2$
$\Rightarrow 2<a<8$