Given, the pole of the line x−5y−7=0 with respect to the circle S≡x2+y2−2x+4y+1=0‌ is ‌P(a,b) ∴ Equation of polar T=0 ‌ax+by−(a+x)+2(b+y)+1=0 ‌(a−1)x+(b+2)y+(2b−a+1)=0 This equation will be same as ‌x−5y−7=0 ‌∴‌‌‌
a−1
1
=‌
b+2
−5
=‌
2b−a+1
−7
‌⇒‌‌a=0‌ and ‌b=3 ‌∴‌‌P(a,b)=P(0,3) Center of circle S=0 is C(1,−2) ‌PC=√(1−0)2+(−2−3)2=√26 ‌√a3+b3−1=√0+27−1=√26 Hence, ‌‌PC=√a3+b3−1
