NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 9: Circles - Exercise 9.2Question 6
Class 10 - Mathematics - CHAPTER 9: Circles - Exercise 9.2
Question. 6
If the angle between two tangents drawn from a point \(P\) to a circle of radius \(a\) and centre \(O\) is \(60^\circ\), then \(OP = a\sqrt{3}\). State True/False and justify.
Answer:
False.
Detailed Answer with Explanation:
Step 1: Let the angle between the tangents be \(\theta = 60^\circ\).
Step 2: Formula for the distance of the external point \(P\) from the centre \(O\) is:
\( OP = \dfrac{a}{\sin(\theta/2)} \)
where \(a\) is the radius of the circle.
Step 3: Substitute the values:
\( OP = \dfrac{a}{\sin(60^\circ / 2)} = \dfrac{a}{\sin 30^\circ} \)
Step 4: We know \(\sin 30^\circ = 1/2\).
So, \( OP = \dfrac{a}{1/2} = 2a \).
Step 5: The given statement says \( OP = a\sqrt{3} \), but our calculation gives \( OP = 2a \).
Final Answer: The statement is False.