Question

If the direct common tangents of the circles $x^2+y^2=1$ and $(x-5{)}^2+y^2=4$ intersect each others at point $D$ and the transverse common tangents to them intersect each other at $T$, then $DT$ is equal to

• A. $5$ units
• B. $\frac{10}{3}$ units
• C. $\frac{20}{3}$ units
• D. $7$ units

Answer 1

The correct option is C $\frac{20}{3}$ units

$\begin{array}{c}\frac{C_{1}T}{C_{2}T}=\frac{1}{2}\\ \left(\frac{1\times 5+2\times 0}{1+2},\frac{1\times 0+2\times 0}{1+2}\right)\\ T\equiv \left(\frac{5}{3},0\right)\\ \frac{C_{1}D}{C_{2}D}=\frac{1}{2}\\ \left(\frac{1\times 5-2\times 0}{1-2},\frac{1\times 0-2\times 0}{1-2}\right)\\ D\equiv \left(-5,0\right)\\ Now,\\ DT\equiv 5+\frac{5}{3}\\ \equiv \frac{20}{3}\\ Hence,optionCisthecorrectanswer.\end{array}$