Question

The triangle \( PQR \) is inscribed in the circle \[ x^2 + y^2 = 25. \] If \( Q = (3,4) \) and \( R = (-4,3) \), then \( \angle QPR \) is:

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)

Hint:

To find the angle between two lines, use the formula for the tangent of the angle between two slopes. For points on a circle, the angle subtended at the center helps in determining angles inside the triangle.

Answer 1

The Correct Option is C

Step 1: Identify the center and radius of the circle 
The given equation of the circle is: \[ x^2 + y^2 = 25. \] This represents a circle centered at \( (0,0) \) with radius \( 5 \). 

Step 2: Finding slopes of \( OQ \) and \( OR \) 
The points \( Q(3,4) \) and \( R(-4,3) \) lie on the circle. The slopes of the lines joining them to the origin (center of the circle) are: \[ \text{Slope of } OQ = \frac{4 - 0}{3 - 0} = \frac{4}{3}, \] \[ \text{Slope of } OR = \frac{3 - 0}{-4 - 0} = -\frac{3}{4}. \] 

Step 3: Find angle between \( OQ \) and \( OR \) 
Using the angle formula between two lines: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|, \] where \( m_1 = \frac{4}{3} \) and \( m_2 = -\frac{3}{4} \), \[ \tan \theta = \left| \frac{\frac{4}{3} + \frac{3}{4}}{1 - \left( \frac{4}{3} \times \frac{3}{4} \right)} \right|. \] 

Step 4: Compute value 
\[ \tan \theta = \left| \frac{\frac{16}{12} + \frac{9}{12}}{1 - \frac{12}{12}} \right| = \left| \frac{\frac{25}{12}}{0} \right|. \] \[ \theta = \frac{\pi}{4}. \] 

Step 5: Conclusion 
Thus, the final answer is: \[ \boxed{\frac{\pi}{4}}. \]