Question

Find the locus of the centre of a circle of radius $r$ touching externally a circle of radius $R$.

Answer 1

Let circle having radius $R$ be centered at $(x_1,y_1)$

Let $(h,k)$ be the center of the circle with radius $r$.

Since they touch externally

$C_1{C}_2=r_1+r_2$

$⟹\sqrt{(h–x_1{)}^2+(k–y_1{)}^2}=r+R$

Locus of the above equation is

$(x–x_1{)}^2+(y–y_1{)}^2=(r+R{)}^2$

Which is a circle with center $(x_1,y_1)$ and radius $r+R$.