Circles
Circle
nta_abhyas_2025
Grade 11
Question:
The angle between the chords of the circle $x^2 + y^2 = 100$, which passes through the point $(7,1)$ and also divides the circumference of the circle into two arcs whose lengths are in the ratio $2 : 1$, is equal to
Step-by-Step Solution
Key Concept: A chord subtending a specific angle at the centre has a fixed distance from the centre; the product of slopes determines perpendicularity.
Let chord $AB$ subtend angle $\theta$ at centre $O(0,0)$. Given $\theta : 2\theta = 30°$, so $\theta = 120° = \angle AOB$. With distance from $O$ to $AB$ equal to $h$, we have $\cos 60° = \frac{h}{r} = \frac{1}{2}$, giving $h = 5$. The chord $AB$ has equation $mz - y - 1 - 7m = 0$ with distance 5 from origin: $\frac{|1-7m|}{\sqrt{1+m^2}} = 5$. Solving: $24m^2 - 14m - 24 = 0$, so $m_1 m_2 = -1$. Therefore the chords are perpendicular.
Correct Answer: 3