If the chord of contact of tangents from a po | vedclass

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(C) Let $C_1: x^2+y^2=r_1^2$,$C_2: x^2+y^2=r_2^2$,and $C_3: x^2+y^2=r_3^2$. Let $P(x_1, y_1)$ be a point on the circle $C_1$,so $x_1^2+y_1^2=r_1^2$. T

Vedclass · Accepted Answer

$GP$

Vedclass · Answer

(C) Let $C_1: x^2+y^2=r_1^2$,$C_2: x^2+y^2=r_2^2$,and $C_3: x^2+y^2=r_3^2$. Let $P(x_1, y_1)$ be a point on the circle $C_1$,so $x_1^2+y_1^2=r_1^2$. The equation of the chord of contact of tangents from $P$ to the circle $C_2$ is given by $T=0$,which is $x x_1+y y_1-r_2^2=0$. Since this line touches the circle $C_3$,the perpendicular distance from the origin $(0,0)$ to the line must be equal to the radius $r_3$. Thus,$\frac{|0 \cdot x_1+0 \cdot y_1-r_2^2|}{\sqrt{x_1^2+y_1^2}}=r_3$. Substituting $x_1^2+y_1^2=r_1^2$,we get $\frac{r_2^2}{\sqrt{r_1^2}}=r_3$,which simplifies to $r_2^2=r_1 r_3$. Therefore,$r_1, r_2, r_3$ are in $GP$.

If the chord of contact of tangents from a point on the circle $x^2+y^2=r_1^2$ to the circle $x^2+y^2=r_2^2$ touches the circle $x^2+y^2=r_3^2$,then $r_1, r_2, r_3$ are in:

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