Two circles with radii ' r1 ' and ' r2 ' r1r2 ≥ 2, touch each other externally. If ' θ ' be the angle between the direct common tangents, then

Question

Two circles with radii ' r1 ' and ' r2 ' r1>r2 ≥ 2, touch each other externally. If ' θ ' be the angle between the direct common tangents, then (A) θ= sin -1((r1+r2/r1-r2)) (B) θ=2 sin -1((r1-r2/r1+r2)) (C) θ= sin -1((r1-r2/r1+r2)) (D) None of these

Tardigrade Admin · Accepted Answer

sin α=(r1-r2/r1+r2) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/1ae57c02c625fe6f1dd7a2fac333ac0d-.png" /> ⇒ θ=2 sin -1(( r 1- r 2/ r 1+ r 2))

Q. Two circles with radii ' $r_{1}$ ' and ' $r_{2}$ ' $r_{1} > r_{2} \geq 2$ , touch each other externally. If ' $θ$ ' be the angle between the direct common tangents, then

$θ = sin^{- 1} (\frac{r _{1} + r _{2}}{r _{1} - r _{2}})$

$θ = 2 sin^{- 1} (\frac{r _{1} - r _{2}}{r _{1} + r _{2}})$

$θ = sin^{- 1} (\frac{r _{1} - r _{2}}{r _{1} + r _{2}})$

None of these

$sin α = \frac{r _{1} - r _{2}}{r _{1} + r _{2}}$

$\Rightarrow θ = 2 sin^{- 1} (\frac{r _{1} - r _{2}}{r _{1} + r _{2}})$

Q. Two circles with radii ' r1​ ' and ' r2​ ' r1​>r2​≥2, touch each other externally. If ' θ ' be the angle between the direct common tangents, then

θ=sin−1(r1​−r2​r1​+r2​​)

θ=2sin−1(r1​+r2​r1​−r2​​)

θ=sin−1(r1​+r2​r1​−r2​​)