Maths · Circle, conic sections: A standard form of equations of a circle, the general form of the equation of a circle, its radius and centre
Chord of the circle passes through the point (7,1) and subtends an angle of at t
Chord \( A B \) of the circle \( x^{2}+y^{2}=100 \) passes through the point (7,1) and subtends an angle of \( 60^{\circ} \) at the circumference of the circle. If \( m_{1} \) and \( m_{2} \) are the slopes of two such chords then the value of \( m_{1} m_{2}, \) is
- A. -1
- B. 1
- C. \( \frac{7}{12} \)
- D. - 3
Step-by-step solution
Given circle radius 10. Angle subtended at circumference 60°, so central angle = 120°, chord length = 2×10×sin60° = 10√3. Chord passes through (7,1). Line: y-1 = m(x-7). Distance from center (0,0) to line = |1-7m|/√(m²+1). Chord length = 2√(r² - d²) = 10√3 implies d=5. So |1-7m| = 5√(m²+1). Squaring: (1-7m)² = 25(m²+1) → 24m² -14m -24=0 → 12m² -7m -12=0. Product of roots = -12/12 = -1.
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