Question

Two fixed circles are in the same plane. If P is a point in that plane and the tangents from P to the two circles are equal then the locus of P

• A. Is a circle
• B. Is a straight line perpendicular to the line of centres
• C. Can not be determined
• D. None of these

Answer 1

The correct option is B Is a straight line perpendicular to the line of centres

Let the general equations of two circles in the same plane are
x2+y2+2g1x+2f1y+c1=0x2+y2+2g1x+2f1y+c1=0 ...(i)
x2+y2+2g2x+2f2y+c2=0x2+y2+2g2x+2f2y+c2=0 ...(ii)
If the lengths of the tangents from a variable point P(h, k) to the above two circles are equal, then
h2+k2+2g1h+2f1k+c1−−−−−−−−−−−−−−−−−−−−−√=h2+k2+2g2h+2f2k+c2−−−−−−−−−−−−−−−−−−−−−√h2+k2+2g1h+2f1k+c1=h2+k2+2g2h+2f2k+c2
⟹2(g1−g2)h+2(f1−f2)k+(c1−c2)=0⟹2(g1−g2)h+2(f1−f2)k+(c1−c2)=0
Clearly, it is a equation of the straight line perpendicular to line joining the centres (−g1,f1)(−g1,f1) and (−g2,−f2)(−g2,−f2) of the above two circles, i.e.,
(y+f1)=(f1−f2)(g1−g2)(x−g1)