Circles
Circle
star_batch_jee_advanced_2025
Grade None
Question:
Considering the circles, $x^2 + y^2 = 25$ and $x^2 + y^2 = 9$. From the point $A(0, 5)$ two segments are drawn touching the inner circle at the points $B$ and $C$ while intersecting the outer circle at the points $D$ and $E$. If $O$ is the centre of both the circles then the length of the segment $OF$ that is perpendicular to $DE$, is:
Step-by-Step Solution
Key Concept: Use complementary angle identities and the sine-cosine relationship to find slopes from the given angle condition.
From $\sin \theta = \frac{3}{5}$, the slopes of chords $AB$ and $AE$ are $\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta$ and $\tan\left(\frac{\pi}{2} + \theta\right) = -\cot \theta$, which equal $\pm\frac{4}{3}$ respectively.
Correct Answer: 1