1.

The lengths of the tangents from any point on the circle `15x^(2)+15y^(2)-48x+64y=0` to the two circles `5x^(2)+5y^(2)-24x+32y+75=0` `5x^(2)+5y^(2)-48x+64y=0` are in the ratioA. `1:2`B. `2:3`C. `3:4`D. none of these

Answer» Correct Answer - A
Let P(h, k) be a point on the circle
`15x^(2)+15y^(2)-48x+64y=0`.
Then, `15h^(2)+15k^(2)-48h+64k=0`
`:. H^(2)+k^(2)=(48)/(15)h-(64)/(15)k " " ...(i)`
Let `PT_(1)` and `PT_(2)` be the lengths of the tangents from P(h, k) to `5x^(2)+5y^(2)-24x+32y+75=0`
and , `5x^(2)+5y^(2)-48x+64y+300=0` respectively. Then,
`PT_(1)=sqrt(h^(2)+k^(2)-(24)/(5)h+(32)/(5)k+15)`
and, `PT_(2)=sqrt(h^(2)+k^(2)-(48)/(5)h+(64)/(5)k+15) " " `
`rArr PT_(1)=sqrt((48)/(15)h -(64)/(15)k-(24)/(5)h+(32)/(5)k+15) " " `[Using (i)]
`rArr PT_(1)=sqrt((32k)/(15)-(24)/(15)h+15)`
and, `PT_(2)=sqrt((48)/(15)h-(64)/(15)h-(64)/(15)k-(48)/(5)h+(64)/(5)k+60)" " `[Using (i)]
`rArr PT_(2)=sqrt(-(96)/(15)h+(128)/(15)k+60)`
`rArr = 2 sqrt(-(24)/(15)h+(32)/(15)k+15)=2 PT_(1)`
`:. PT_(1):PT_(2)=1:2`


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