Given circle is x2+y2−2x+4y+3=0 So, center=(−g,−f)=(‌
−2
2
,‌
4
2
) =(−1,2)‌ and ‌c=3 Also, the equation of the polar of a point(x1,y1)w.r.t the circlex2+y2+2gx+2fy+c=0is ‌xx1+yy1+g(x+x1)+f(y+y1)+c=0 ‌⇒xx1+yy1−(x+x1)+2(y+y1)+3=0 ‌⇒xx1+yy1−x−x1+2y+2y1+3=0 ‌⇒(x1−1)x+(y1+2)y−x1+2y1+3=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(i) Given, polar equation is 2x−3y+1=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(ii) By comparing the coefficients of two equations, we get ‌
x1−1
2
‌=‌
y1+2
−3
‌=‌
−x1+2y1+3
1
⇒‌‌‌
x1−1
2
‌=‌
y1+2
−3
‌ and ‌ ‌
x1−1
2
‌=‌
−x1+2y1+3
1
⇒−3x1+3‌=2y1+4‌ and ‌x1−1 ‌=−2x1+4y1+6 ⇒3x1+2y1‌=−1‌ and ‌3x1−4y1−7=0 By solving these two equations, we getx1=‌