A common tangent to `9x^(2) - 16y^(2) = 144` and `x^(2) + y^(2) = 9` is

Options

A. `y=(3)/(sqrt7)x+(15)/(7)`
B. `y=3sqrt((2)/(7))x+(15)/(7)`
C. `y=2sqrt((3)/(7))x+15sqrt7`
D. None of the above

Answer

B

Explanation

Let y=mx+c be a common tangent to `9x^(2)-16y^(2)=144 and x^(2)+y^(2)=9`. Since `y=mx+c` is a tangent to `9x^(2)-16y^(2)=144`
`9x^(2)-16y^(2)=144`
`therefore c^(2)=a^(2)m^(2)-b^(2)Rightarrow c^(2)=16m^(2)-9.....(i)`
`therefore (c)/(sqrt(m^(2)+1))=3 Rightarrow c^(2)=9(1+m^(2)).....(ii)`
From Eqs. (i) and (ii), we get
`=16m^(2)-9=9+9m^(2)`
`Rightarrow m=pm 3sqrt((2)/(sqrt7))`
On putting the value of m in Eq( (ii), we get
`c=pm sqrt((2)/(7))`
Hence, `y=3sqrt(2)/(7)x+(15)/(7)` is a common tangent.