Question

The circles $x^2+y^2-5x+6y+15=0$ and $x^2+y^2-2x+6y+6=0$ touch each other

• A. Internally
• B. Externally
• C. Do not say anything
• D. None of these

Answer 1

The correct option is A Internally

The equation of given circles are

$x^2+y^2-5x+6y+15=0$ ...... $\left(i\right)\to C_1$

and $x^2+y^2-2x+6y+6=0$ ...... $\left(ii\right)\to C_2$

The center and radii of given circles are
$C_1(\frac{5}{2},-3),$ $C_2(1,-3)$ and $r_1=\sqrt{\frac{25}{4}+9-15}=\sqrt{\frac{25-24}{4}}=\frac{1}{2},$ $r_2=\sqrt{1+9-6}=2$,

Now, $C_1{C}_2=\sqrt{{(\frac{5}{2}-1)}^2+{(-3+3)}^2}$
$=\sqrt{{\left(\frac{3}{2}\right)}^2+0}=\frac{3}{2}$
and $|r_1-r_2|=2-\frac{1}{2}=\frac{3}{2}$
$\because |r_1-r_2|=C_1{C}_2$
Hence, the two circles touch each other internally.