Circles
Circle
nta_abhyas_2025
Grade 11
Question:
From a point $P(3, 3)$ on the circle $x^2 + y^2 = 18$, two chords $PQ$ and $PR$ each of 2 units length are drawn on this circle. The value of $\cos(\angle QPR)$ is equal to
Step-by-Step Solution
Key Concept: The double angle formula $\cos 2\theta = 2\cos^2\theta - 1$ converts a single angle cosine value into a double angle cosine value
Using the Law of Cosines in triangle $OPR$, we find $\cos\theta = \frac{1}{3}$. To find $\cos(\angle QPR) = \cos 2\theta$, we apply the double angle formula: $\cos 2\theta = 2\cos^2\theta - 1 = 2 \times \left(\frac{1}{3}\right)^2 - 1 = \frac{2}{9} - 1 = -\frac{7}{9}$.
Correct Answer: 2