Question

Locus of the center of circle of radius $4$ which touches the circle $x^2+y^2-4x+2y-4=0$ externally is

• A. $x^2+y^2-4x+2y-44=0$
• B. $x^2+y^2+4x+2y-44=0$
• C. $x^2+y^2-4x-2y-44=0$
• D. None of these

Answer 1

The correct option is A $x^2+y^2-4x+2y-44=0$

Let the center of the circle $S_1$ be $C_1(x_1,y_1)$

Its radius $=r_1=4$.

Given circle is $S_2\equiv x^2+y^2-4x+2y-4=0$

Its center is $C_2(2,-1)$ and radius $=r_2=\sqrt{4+1+4}=3$

Alos, $d=$ distance between the centers $=\sqrt{{(x_1-2)}^2+{(y_1+1)}^2}$

Since the two circles touch each other ecxternally

$\therefore d=r_1+r_2\Rightarrow \sqrt{{(x_1-2)}^2+{(y_1+1)}^2}=4+3\Rightarrow {x_1}^2+{y_1}^2-4x_1+2y_1+5=49$

$\therefore$ Locus of $(x_1,y_1)$ is

$x^2+y^2-4x+2y-44=0$