Chapter 9: Q18. (page 337)

Circles $P$ and $Q$ have radii 6 and 2 and are tangent to each other. Find length of their common external tangent $\bar{A B .}$

Short Answer

Expert verified

The required length is,

$A B = 4 \sqrt{5}$ .

Step by step solution

Step 1. Given information.

Circles $P$ and $Q$ have radii 6 and 2 and are tangent to each other.

Given figure,

Step 2. Concept used.

There are two circles whose radius is 6 and 2 respectively.

Joint $A P$ and $B Q$ .

Note that $A P$ and $B Q$ are radius, then

$\begin{array}{l} A P = 6 \\ B Q = 2 \end{array}$

Step 3. Draw the construction.

Joint $P Q$ and joint $O B$ and make a parallel line to $P Q$ . Then $O P Q B$ will be a rectangle.

In the rectangle $O P Q B$ opposite sides are equal.

$\begin{array}{l} O B = P Q (= 4) \\ O P = Q B (= 2) ................. (1) \end{array}$

Then,

$\begin{matrix} A P = A O + O P \\ 6 = A O + 2 \\ A O = 6 - 2 \\ A O = 4............. (1) \end{matrix}$

The distance $P Q$ is,

$\begin{matrix} P Q = r a d i u s 1 + r a d i u s 2 \\ = 6 + 2 \\ = 8 \end{matrix}$

From (1), $O B = P Q$ and $P Q = 8$

This implies that $O B = 8$

Note that $Δ A O B$ is a right-angled triangle.

Step 4. Use Pythagoras theorem.

$\begin{matrix} A B^{2} = O A^{2} + O B^{2} \\ = 4^{2} + 8^{2} \\ = 16 + 64 \\ = 80 \end{matrix}$

Take positive square root to get,

$\begin{matrix} A B = \sqrt{80} \\ = 4 \sqrt{5} \end{matrix}$

Therefore, the required length is, $A B = 4 \sqrt{5}$ .

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Circles $P$ and $Q$ have radii 6 and 2 and are tangent to each other. Find length of their common external tangent $\bar{A B .}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. Draw the construction.

Step 4. Use Pythagoras theorem.

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Circles Pand Qhave radii 6 and 2 and are tangent to each other. Find length of their common external tangent AB.¯

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. Draw the construction.

Step 4. Use Pythagoras theorem.

One App. One Place for Learning.

Most popular questions from this chapter

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Circles $P$ and $Q$ have radii 6 and 2 and are tangent to each other. Find length of their common external tangent $\bar{A B .}$