The lengths of the tangents from any point on  the circle 15x2+15y2−48x+64y=0 to the circles5x2+5y2−24x+32y+75=05x2+5y2−48x+64y+300=0 are in the ratio

Let P (h, k) be a point on the circle15x2+15y2−48x+64y=0Then 15h2+15k2−48h+64k=0∴ h2+k2=4815h−6415k             …(i)Let PT1 and PT2 be the lengths of the tangents from P (h, k) to 5x2+5y2−24x+32y+75=0and, 5x2+5y2−48x+64y+300=0 respectively. Then,PT1=h2+k2−245h+325k+15and, PT2=h2+k2−485h+645k+60⇒ PT1=4815h−6415k−245h+325k+15       [Using (i)] ⇒ PT1=32k15−2415h+15and, PT2=4815h−6415k−485h+645k+60            [Using (i)] ⇒ PT2=−9615h+12815k+60⇒ PT2=2−2415h+3215k+15=2PT1∴ PT1:PT2=1:2