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

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:

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:

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:

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: