### Circles

goals
If the polar of the point $(x_{1}, y_{1})$ with respect to the circle $x^{2} + y^{2} = a^{2}$ touches the circle $(x - a)^{2} + y^{2} = a^{2}$, prove that the locus of the point $(x_{1}, y_{1})$ is $y^{2} + 2ax - a^{2} = 0$.
Let $A\equiv\left(a,0\right)$ and $B\equiv\left(-a,0\right)$ be two fixed points for all $a\in \left(-\infty,0\right)$ and P moves on a plane such that $PA=n.PB$ where $n\neq 0$
On the basis of the above information, answer the following questions
Equation of a circle is  $S={x}^{2}+{y}^{2}+2gx+2fy+c$
Its notation is ${S}_{1}={x}_{1}x+{y}_{1}y+g\left(x+{x}_{1}\right)+f\left({y}_{1}+y\right)+c$
${S}_{11}={x}_{1}^{2}+{y}_{1}^{2}+2g{x}_{1}+2f{y}_{1}+c$
$\left(i\right)$Location of $P\left({x}_{1},{y}_{1}\right):$
$P$ lies inside the circle $S=0$, if ${S}_{11}<0$
$P$ lies outside the circle $S=0$ if ${S}_{11}>0$
$P$ lies on the circle $S=0,$ if ${S}_{11}=0$
$\left(ii\right)$Tangent at $P\left({x}_{1},{y}_{1}\right)$ on the circle $S=0,$ is ${S}_{1}=0$
$\left(iii\right)$ Length of the tangent from the point $\left({x}_{1},{y}_{1}\right)$ to the circle $S=0$ is $\sqrt{{S}_{11}}$
$\left(iv\right)$Pair of tangents $PQ,PR$ from $P\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}^{2}={S}_{11}S$
$\left(v\right)$Chord of contact $QR$ of tangents from $P\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}=0$
$\left(vi\right)$Chord of $S=0$ with midpoint $\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}={S}_{11}$
Based on the above information,answer the following questions:
Equation of a circle is  $S={x}^{2}+{y}^{2}+2gx+2fy+c$
Its notation is ${S}_{1}={x}_{1}x+{y}_{1}y+g\left(x+{x}_{1}\right)+f\left({y}_{1}+y\right)+c$
${S}_{11}={x}_{1}^{2}+{y}_{1}^{2}+2g{x}_{1}+2f{y}_{1}+c$
$\left(i\right)$Location of $P\left({x}_{1},{y}_{1}\right):$
$P$ lies inside the circle $S=0$, if ${S}_{11}<0$
$P$ lies outside the circle $S=0$ if ${S}_{11}>0$
$P$ lies on the circle $S=0,$ if ${S}_{11}=0$
$\left(ii\right)$Tangent at $P\left({x}_{1},{y}_{1}\right)$ on the circle $S=0,$ is ${S}_{1}=0$
$\left(iii\right)$ Length of the tangent from the point $\left({x}_{1},{y}_{1}\right)$ to the circle $S=0$ is $\sqrt{{S}_{11}}$
$\left(iv\right)$Pair of tangents $PQ,PR$ from $P\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}^{2}={S}_{11}S$
$\left(v\right)$Chord of contact $QR$ of tangents from $P\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}=0$
$\left(vi\right)$Chord of $S=0$ with midpoint $\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}={S}_{11}$
Based on the above information,answer the following questions:
Equation of a circle is  $S={x}^{2}+{y}^{2}+2gx+2fy+c$
Its notation is ${S}_{1}={x}_{1}x+{y}_{1}y+g\left(x+{x}_{1}\right)+f\left({y}_{1}+y\right)+c$
${S}_{11}={x}_{1}^{2}+{y}_{1}^{2}+2g{x}_{1}+2f{y}_{1}+c$
$\left(i\right)$Location of $P\left({x}_{1},{y}_{1}\right):$
$P$ lies inside the circle $S=0$, if ${S}_{11}<0$
$P$ lies outside the circle $S=0$ if ${S}_{11}>0$
$P$ lies on the circle $S=0,$ if ${S}_{11}=0$
$\left(ii\right)$Tangent at $P\left({x}_{1},{y}_{1}\right)$ on the circle $S=0,$ is ${S}_{1}=0$
$\left(iii\right)$ Length of the tangent from the point $\left({x}_{1},{y}_{1}\right)$ to the circle $S=0$ is $\sqrt{{S}_{11}}$
$\left(iv\right)$Pair of tangents $PQ,PR$ from $P\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}^{2}={S}_{11}S$
$\left(v\right)$Chord of contact $QR$ of tangents from $P\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}=0$
$\left(vi\right)$Chord of $S=0$ with midpoint $\left({x}_{1},{y}_{1}\right)$ is ${S}_{1}={S}_{11}$
Based on the above information,answer the following questions: