Question

The centers of two circles with radii 8 cm and 3 cm are 13 cm apart A direct common tangent touches the circles at A and B respectively then the length of AB is

• A. $8$ cm
• B. $10$ cm
• C. $12$ cm
• D. none of this

Answer 1

The correct option is C $12$ cm

$\Rightarrow$ Let $O_1$ and $O_2$ be the center of ciecle $X$ and circle $Y$, respectively, as shown in the diagram.

$\Rightarrow$ Radius of circle X i.e $O_{1}A=8cm.$

$\Rightarrow$ Radius of circle Y i.e $O_{2}B=3cm.$

$\Rightarrow$ Distance between the centers of the two circles i.e $O_1{O}_2=13cm.$

$\Rightarrow$ Length of a common tangent $AB$

$\Rightarrow$ Construct line segment $P{O}_2$, which is parallel to $AB$.

$\Rightarrow$ $P{O}_1=O_{1}A-O_{2}B=8-3=5cm.$

$\Rightarrow$ $△P{O}_1{O}_2$ is right angled triangle.

$\therefore$ $P{O}_2=\sqrt{(O_1{O}_2{)}^2-(O_{1}P{)}^2}$ [ By using Pythagoras theorem ]

$\therefore$ $P{O}_2=\sqrt{{13}^2-5^2}=\sqrt{169-25}=\sqrt{144}$

$\therefore$ $P{O}_2=12cm$

$\Rightarrow$ Hence, length of common tangent $AB=P{O}_2=12cm$