Question

The range of values of $\lambda$ for which the circles $x^2+y^2=4$ and $x^2+y^2-4\lambda x+9=0$ have two common tangents is

• A. $[-13/8,13/8]$
• B. $>13/8Or<-13/8$
• C. $(1,13/8)$
• D. none of these

Answer 1

The correct option is B $>13/8Or<-13/8$

$C_1=(0,0),r_1=2$

$C_2=(2\lambda ,0),r_2=\sqrt{4{\lambda }^2-9}$

They must have 2 common tangents

$⟹|r_1–r_2|<C_1{C}_2<r_1+r_2$

$C_1{C}_2<r_1+r_2$

$2\lambda <2+\sqrt{4{\lambda }^2-9}$

$⟹4{\lambda }^2+4–8\lambda <4{\lambda }^2-9$

$⟹8\lambda >13⟹\lambda >\frac{13}{8}$............(1).

$C_1{C}_2>|r_1-r_2|$

$C_1{C}_2^2>(r_1–r_2{)}^2$

$4\lambda >4+4{\lambda }^2–9–4\sqrt{4{\lambda }^2-9}$

$4\sqrt{4{\lambda }^2-9}>-5$

$4{\lambda }^2–9>\frac{25}{16}$

$4{\lambda }^2=\frac{169}{16}$

${\lambda }^2–(\frac{13}{8}{)}^2>0$

$⟹(\lambda +\frac{13}{8})(\lambda -\frac{13}{8})>0$

$⟹\lambda >\frac{-13}{8}$ and $\lambda >\frac{13}{8}$ or $\lambda <\frac{-13}{8}$ and $\lambda <\frac{13}{8}$.............(2).

Combining (1) and (2) we get,

$\lambda >\frac{13}{8}$ or $\lambda <-\frac{13}{8}$