NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 9: Circles - Exercise 9.2Question 5
Class 10 - Mathematics - CHAPTER 9: Circles - Exercise 9.2
Question. 5
In Fig. 9.13, \(AB\) and \(CD\) are common tangents to two circles of unequal radii. Prove that \(AB=CD\).
Answer:
\(AB=CD\)
Detailed Answer with Explanation:
Step 1: Let the two common tangents \(AB\) and \(CD\) meet at a point \(P\) outside the circles.
Step 2: From geometry, we know that the lengths of tangents drawn from the same external point to a circle are equal.
Step 3: Apply this property to the left circle:
- \(PA\) and \(PC\) are tangents from point \(P\).
- Therefore, \(PA = PC\).
Step 4: Apply the same property to the right circle:
- \(PB\) and \(PD\) are tangents from point \(P\).
- Therefore, \(PB = PD\).
Step 5: Look at line \(PAB\):
- \(AB = PA - PB\).
Step 6: Now look at line \(PCD\):
- \(CD = PC - PD\).
Step 7: Since \(PA = PC\) and \(PB = PD\), subtracting gives:
\[ AB = PA - PB = PC - PD = CD \]
Final Result: Hence, we have proved that \(AB = CD\).