The lengths of the tangent drawn from any point on the circle 15 x²+15 y²-48 x+64 y=0 to the two circles 5 x²+5 y²-24 x+32 y+75=0 and 5 x²+5 y²-48 x+64 y+300=0 are in the ratio of

Step-by-step solution

Let ( P ( h , k ) ) be a point on the circle (15 x²+15 y²-48 x+64 y=0 ) Then the lengths of the tangents from ( P ( h , k ) ) to ( aligned &5 x²+5 y²-24 x+32 y+75=0 &5 x²+5 y²-48 x+64 y+300=0 are &P T₁= h²+k²- 24 5 h+ 32 5 k+15 & and P T₂= h²+k²- 48 5 h+ 64 5 k+60 & or P T₁= 48 15 h- 64 15 k- 24 5 h+ 32 5 k+15 = 32 15 k- 24 15 h+15 aligned ) (Since (( h , k ) ) lies on (15 x ²-15 y ²-48 x +64 y =0 ) ( aligned & . h ²+ k ²- 48 15 ~h + 64 15 k =0 ) & and PT ₂= 48 15 ~h - 64 5 k - 48 5 ~h + 64 5 k +60 &= - 96 15 ~h +