Question

The two circles $x^2+y^2=ax$ and$x^2+y^2=c^2(c>0)$ touch each other if

• A. $|a|=c$
• B. $a=2c$
• C. $|a|=2c$
• D. $2|a|=c$

Answer 1

The correct option is A $|a|=c$

The 2 circle $x^2+y^2=ax$ & $x^2+y^2=c^2(c>0)$ touch if:

$1^{st}$ circle:

$x^2+y^2=ax$ ___ (1)

The centre lie at $c_1:(\frac{-a}{2},0)$ & radius $r_1=\left|\frac{a}{2}\right|$

$2^{nd}$ circle: $x^2+y^2=c^2$ ___ (2)

$c_2=(0,0)$ , $r_2=c$

2 circle touch each other if:

$\left|c_1{c}_2\right|=r_1\pm r_2$

$\therefore \left|{(-\frac{a}{2})}^2\right|={(\pm \frac{a}{2}\pm c)}^2$

$\frac{a^2}{4}=\frac{a^2}{4}+c^2\pm \left|a\right|c$

$\left|a\right|c=c^2$

$\left|a\right|=c$ (Option A)