Question

If $ax+by+c=0$ is the polar of $(1,1)$ for the circle $x^2+y^2-2x+2y+1=0$ and $H.C.F.$ of $b,c$ is equal to $1$, then the value of $a^2+b^2+c^2$ is

• A. $0$
• B. $3$
• C. $5$
• D. $15$

Answer 1

The correct option is C $5$

Given circle is $x^2+y^2-2x+2y+1=0$
Now, the equation of polar is
$T=0\Rightarrow x\left(1\right)+y\left(1\right)-(x+1)+(y+1)+1=0\Rightarrow 2y+1=0\Rightarrow 0\cdot x+2\cdot y+1=0$
Given equation of polar is $ax+by+c=0$, so
$a=0$
Now,
$\frac{b}{2}=\frac{c}{1}=k\Rightarrow b=2k,c=k$
As the $H.C.F.$ of $b,c$ is $1$, so
$b=2,c=1\therefore a^2+b^2+c^2=5$