Question

If two circles $x^2+y^2+2gx+2fy=0$ and $x^2+y^2+2g_{1}x+2f_{1}y=0$ touch each other, then

• A. $f_{1}g=f{g}_1$
• B. $f{f}_1=g{g}_1$
• C. $f^2+g^2=f_1^2+g_1^2$
• D. None of these

Answer 1

The correct option is A $f_{1}g=f{g}_1$

The cordinates of centre and radius is calculated as
for first circle: centre $c_1(-g,-f)$ and radius $r_1=\sqrt{g^2+f^2}$
for second circle: centre $c_2(-g_1,-f_1)$ and radius $r_2=\sqrt{g_1^2+f_1^2}$
Both circles touch each other when
$c_1{c}_2=r_1+r_2$
$\Rightarrow \sqrt{(g-g_1{)}^2+(f-f_1{)}^2}=\sqrt{g^2+f^2}+\sqrt{g_1^2+f_1^2}$
squaring both side
$\Rightarrow -2g{g}_1-2f{f}_1=2\sqrt{(g^2+f^2)(g_1^2+f_1^2)}$
again squaring both side
$\Rightarrow (g{g}_1+f{f}_1{)}^2=(g^2+f^2\left)\right(g_1^2+f_1^2)$
$\Rightarrow 2g{g}_{1}f{f}_1=g^2{f}_1^2+f^2{g}_1^2$
$\Rightarrow (f_{1}g-f{g}_1{)}^2=0$
$\Rightarrow f_{1}g=f{g}_1$